Skew mirrors, methods of use, and methods of manufacture

ABSTRACT

An optical reflective device referred to as a skew mirror, having a reflective axis that need not be constrained to surface normal, is described. Examples of skew mirrors are configured to reflect light about substantially constant reflective axes across a relatively wide range of wavelengths. In some examples, a skew mirror has substantially constant reflective axes across a relatively wide range of angles of incidence. Exemplary methods for making and using skew mirrors are also disclosed. Skew mirrors include a grating structure, which in some examples comprises a hologram.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application claims priority from co-pending U.S. application Ser. No. 15/517,159, titled “SKEW MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” which entered the US National Stage on 5 Apr. 2017 from International Patent Application No. PCT/US2016/048499, filed 24 Aug. 2016 and titled “SKEW MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” and Ser. No. 15/174,938, filed 6 Jun. 2016 and titled “SKEW MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” which claims priority to U.S. Application Nos. 62/209,290, filed 24 Aug. 2015 and titled “MULTIWAVELENGTH DIFFRACTION GRATING MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” and 62/318,917, filed 6 Apr. 2016 and titled “SKEW MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE.” The above applications are incorporated herein by reference, in their entireties.

BACKGROUND

Conventional dielectric mirrors are produced by coating a surface (typically glass) with layers of materials that differ from each other in their electric permittivity. The layers of materials are typically arranged so that Fresnel reflections from layer boundaries reinforce constructively, producing large net reflectivity. Broadband dielectric mirrors can be designed by ensuring that this condition obtains over a relatively broad specified range of wavelengths and incidence angles. However, because the layers are deposited on a surface, the reflective axis of a dielectric mirror is necessarily coincident with surface normal, i.e. the reflective axis is perpendicular to the mirror surface. Because of this constraint on the reflective axis, a dielectric mirror is entirely inadequate for some purposes. Moreover, glass dielectric mirrors tend to be relatively heavy, making them suboptimal or inappropriate for applications requiring a relatively lightweight reflective component.

Conversely, conventional grating structures can reflect light about a reflective axis that differs from surface normal of the medium in which the grating structure resides. However, for a given angle of incidence, angles of reflection for conventional grating structures typically co-vary with wavelength of incident light. Thus, using a conventional grating structure to reflect light avoids the constraint inherent in dielectric mirrors that reflective axes must coincide with surface normal. However, where a constant reflective axis is required, a conventional grating structure is typically limited to a single wavelength or very narrow range of wavelengths for a given angle of incidence. Similarly, a conventional grating structure is limited to a single angle of incidence or very narrow range of incidence angles in order to reflect light of a specified wavelength about a constant reflective axis. Accordingly, a conventional grating structure does not have a constant reflective axis over any significant range of wavelengths or angles of incident light.

Accordingly, requirements for a relatively simple device that reflects light about a reflective axis not constrained to surface normal, and whose angle of reflection for a given angle of incidence is substantially constant at multiple wavelengths, are not met by currently available reflective devices comprising either reflective grating structures or dielectric mirrors. A need therefore exists for such a reflective device, and such need may be acute in head mounted display devices.

BRIEF DESCRIPTION OF THE DRAWINGS

The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar elements).

FIG. 1A is a cross-section view of a hologram recorded in a grating medium.

FIG. 1B is a cross-section view of a k-space representation of a single sinusoidal hologram.

FIG. 2A is a cross-section view of a k-space representation of a single sinusoidal hologram.

FIG. 2B cross-section view of a k-space representation of a single sinusoidal hologram.

FIG. 3 is a cross-section real view illustrating reflective properties of a skew mirror in real space, according to an embodiment.

FIG. 4A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 4B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 5A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 5B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 6A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 6B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 6C is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 6D is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 7A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 7B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 8A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 8B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 8C is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 9A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 9B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 10A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

FIG. 10B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 11A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 11B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 12A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 12B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 13 is a cross-section view of a system for making a skew mirror, according to an embodiment.

FIG. 14 is a cross-section view illustrating a method of making a skew mirror, according to an embodiment.

FIG. 15 is a plan view illustrating reflective properties of a skew mirror according to an embodiment.

FIG. 16A is a cross-section view illustrating a system for making a skew mirror, according to an embodiment.

FIG. 16B is a cross-section view illustrating a system for making a skew mirror, according to an embodiment.

DETAILED DESCRIPTION

Embodiments of the present invention include a reflective device comprising a grating medium within which resides a volume hologram or other grating structure. The grating medium, by virtue of the grating structure residing therein, has physical properties that allow it to diffract light about an axis, referred to as a reflective axis, wherein angle of diffraction (henceforth referred to as angle of reflection) varies by less than 1° for multiple wavelengths of light incident upon the grating medium at a given angle of incidence. In some embodiments, the above phenomenon is observed for multiple angles of incidence.

Similarly, embodiments typically have substantially constant reflective axes (i.e., reflective axes have reflective axis angles that vary by less than 1.0 degree) across a range of incidence angles for incident light of a given wavelength, and this phenomenon may be observed with incident light at various wavelengths. In some embodiments, the reflective axes remain substantially constant for every combination of a set of multiple incidence angles and a set of multiple wavelengths

In some embodiments, the grating structure includes a hologram generated by interference between multiple light beams referred to as recording beams. Typically, but not necessarily, the grating structure includes multiple holograms. The multiple holograms may be recorded using recording beams incident upon the grating medium at angles that vary among the multiple holograms (i.e. angle multiplexed), and/or using recording beams whose wavelengths vary among the multiple holograms (i.e. wavelength multiplexed). In some embodiments, the grating structure includes a hologram recorded using two recording beams whose angles of incidence upon the grating medium vary while the hologram is being recorded, and/or whose wavelengths vary while the hologram is being recorded. Embodiments further include a device wherein the reflective axis differs from surface normal of the grating medium by at least 1.0 degree; or at least by 2.0 degrees; or at least by 4.0 degrees; or at least by 9.0 degrees.

k-Space Formalism for Holography

The k-space formalism is a method for analyzing holographic recording and diffraction. In k-space, propagating optical waves and holograms are represented by three dimensional Fourier transforms of their distributions in real space. For example, an infinite collimated monochromatic reference beam can be represented in real space and k-space by equation (1), E _(r)({right arrow over (r)})=A _(r)exp(i {right arrow over (k)} _(r) ·{right arrow over (r)})

E _(r)({right arrow over (k)})=A _(r)δ({right arrow over (k)}−{right arrow over (k)} _(r)),  (1) where E_(r)({right arrow over (r)}) is the optical scalar field distribution at all {right arrow over (r)}={x, y, z} 3D spatial vector locations, and its transform E_(r) ({right arrow over (k)}) is the optical scalar field distribution at all {right arrow over (k)}={k_(x),k_(y),k_(z)} 3D spatial frequency vectors. A_(r) is the scalar complex amplitude of the field; and {right arrow over (k)}_(r) is the wave vector, whose length indicates the spatial frequency of the light waves, and whose direction indicates the direction of propagation. In some embodiments, all beams are composed of light of the same wavelength, so all optical wave vectors must have the same length, i.e., |{right arrow over (k)}_(r)|=k_(n). Thus, all optical propagation vectors must lie on a sphere of radius k_(n)=2πn₀/λ, where n₀ is the average refractive index of the hologram (“bulk index”), and λ is the vacuum wavelength of the light. This construct is known as the k-sphere. In other embodiments, light of multiple wavelengths may be decomposed into a superposition of wave vectors of differing lengths, lying on different k-spheres.

Another important k-space distribution is that of the holograms themselves. Volume phase holograms usually consist of spatial variations of the index of refraction within a grating medium. The index of refraction spatial variations, typically denoted Δn({right arrow over (r)}), can be referred to as index modulation patterns, the k-space distributions of which are typically denoted Δn({right arrow over (k)}). The index modulation pattern created by interference between a first recording beam and a second recording beam is typically proportional to the spatial intensity of the recording interference pattern, as shown in equation (2), Δn({right arrow over (r)})∝|E ₁({right arrow over (r)})+E ₂({right arrow over (r)})² |=|E ₁({right arrow over (r)})|² +|E ₂({right arrow over (r)})|² +E ₁*({right arrow over (r)})E ₂({right arrow over (r)})+E ₁({right arrow over (r)})E ₂*({right arrow over (r)}),  (2) where E₁({right arrow over (r)}) is the spatial distribution of the signal first recording beam field and E₂ ({right arrow over (r)}) is the spatial distribution of the second recording beam field. The unary operator * denotes complex conjugation. The final term in equation (2), E₁ ({right arrow over (r)})E₂*({right arrow over (r)}), maps the incident second recording beam into the diffracted first recording beam. Thus we can write equation (3), E ₁({right arrow over (r)})E ₂*({right arrow over (r)})

E ₁({right arrow over (k)})⊗E ₂({right arrow over (k)}),  (3) where ⊗ is the 3D cross correlation operator. This is to say, the product of one optical field and the complex conjugate of another in the spatial domain becomes a cross correlation of their respective Fourier transforms in the frequency domain.

FIG. 1A illustrates a real space representation of recording a hologram 105 in a grating medium 110 using a second recording beam 115 and a first recording beam 114. The grating medium typically includes a recording layer configured to record interference patterns as holograms. FIG. 1A omits grating medium components other than the recording layer, such as an additional layer that might serve as a substrate or protective layer for the recording layer. The second recording beam 115 and first recording beam 114 are counter-propagating. Each of the second recording beam 115 and first recording beam 114 are typically plane wave beams having the same wavelength as each other, and the first recording beam 114 typically contains no encoded information that is not also present in the second recording beam. Thus the first and second recording beams, which can be referred to as signal and reference beams, are typically substantially identical to each other except for angles at which they are incident upon the recording medium 110.

FIG. 1B illustrates a k-space representation of the first and second recording beams, and the hologram. The hologram illustrated in FIGS. 1A and 1B is a simple Bragg reflection hologram generated with the counter-propagating first recording beam 114 and second recording beam 115, and recorded in recording medium 110. FIG. 1A shows the second recording beam 115 and the first recording beam 114 impinging on opposite sides of the grating medium 110. Optical scalar field distributions at all {right arrow over (r)}={x, y, z} 3D spatial vector locations for each of the second recording beam 115 and the first recording beam 114 can be represented as E₂({right arrow over (r)}) and E₁ ({right arrow over (r)}), respectively. The recording beams 114, 115 form planar interference fringes, which are recorded as a hologram 105 within the grating medium 110. The hologram 105 comprises a sinusoidal refractive index modulation pattern, and can be represented as Δn({right arrow over (r)}). In a counter-propagating configuration, the recorded planar interference fringes have a spacing exactly half that of the (internal) wavelength of the light used to record the hologram.

FIG. 1B shows a k-space representation of the situation illustrated in real space by FIG. 1A. The recording beams are represented in FIG. 1B by point-like k-space distributions lying on opposite sides of the recording k-sphere 170. As illustrated in FIG. 1B, the second recording beam has a k-space distribution 162, and the first recording beam has a k-space distribution 163. The second recording beam k-space distribution 162 can be represented as E₂({right arrow over (k)}) and the first recording beam k-space distribution 163 can be represented as E₁({right arrow over (k)}). Each of the second recording beam k-space distribution 162 and the first recording beam k-space distribution 163 are “point-like.” Second recording beam wave vector 164 and first recording beam wave vector 165, are shown extending from the origin to the second recording beam k-space distribution 162 and first recording beam k-space distribution 163, respectively. The second recording beam wave vector 164 can be represented as E₂({right arrow over (k)}) and the first recording beam wave vector 165 can be represented as E₁({right arrow over (k)}). The hologram itself is represented in FIG. 1B by two conjugate sideband k-space distributions 168, each of which can be represented as Δn({right arrow over (k)}) and referred to as a Δn({right arrow over (k)}) k-space distribution. The two Δn({right arrow over (k)}) k-space distributions 168 have a small, finite size, but are “point-like” in the sense that they are typically several orders of magnitude smaller than their distance to the origin, or other features of FIG. 1B. For instance, if the thickness of grating medium 110 is 200 μm with refractive index 1.5 and the recording beams have a wavelength of 532 nm, then distributions 168 each resemble a sinc function along the k_(z) dimension with size 3.14×10⁴ radians per meter (rad/m) null-to-null. However, their distance from the origin is 3.56×10⁷ rad/m, which is more than 1000 times as large. Unless specified otherwise, all recited wavelengths refer to vacuum wavelengths.

Typically, the hologram constitutes a refractive index distribution that is real-valued in real space. Locations of the two Δn({right arrow over (k)}) k-space distributions 168 of the hologram may be determined mathematically from the cross-correlation operations E₂({right arrow over (k)})⊗E₁({right arrow over (k)}) and E₁({right arrow over (k)})⊗E₂({right arrow over (k)}) respectively, or geometrically from vector differences {right arrow over (K)}_(G+)={right arrow over (k)}₁−{right arrow over (k)}₂ and {right arrow over (K)}_(G−)={right arrow over (k)}₂−{right arrow over (k)}₁, where {right arrow over (K)}_(G+) and {right arrow over (K)}_(G−) are grating vectors from the respective hologram Δn({circumflex over (k)}) k-space distributions to the origin (not shown individually). A grating vector 169, which can be represented as {right arrow over (K)}_(G), comprising both {right arrow over (K)}_(G+) and {right arrow over (K)}_(G−) grating vectors, is shown in FIG. 1B as double headed arrow 169 extending between the second recording beam k-space distribution 162 and the first recording beam k-space distribution 163. Note that by convention, wave vectors are represented by a lowercase “k,” and grating vectors by uppercase “K.”

Once recorded, the hologram may be illuminated by a probe beam to produce a diffracted beam. For purposes of the present disclosure, the diffracted beam can be considered a reflection of the probe beam, which can be referred to as an incident light beam. The probe beam and its reflected beam are angularly bisected by a reflective axis (i.e. the angle of incidence of the probe beam relative to the reflective axis has the same magnitude as the angle of reflection of the reflected beam relative to the reflective axis). The diffraction process can be represented by a set of mathematical and geometric operations in k-space similar to those of the recording process. In the weak diffraction limit, the diffracted light distribution of the diffracted beam is given by equation (4), E _(d)({right arrow over (k)})∝Δn({right arrow over (k)})*E _(p)({right arrow over (k)})|_(|k|=k) _(n) ,  (4)

where E_(d) ({right arrow over (k)}) and E_(p)({right arrow over (k)}) are k-space distributions of the diffracted beam and the probe beam, respectively; and “*” is the 3D convolution operator. The notation “|_(|{right arrow over (k)}|=k) _(n) ” indicates that the preceding expression is evaluated only where |{right arrow over (k)}|=k_(n), i.e., where the result lies on the k-sphere. The convolution Δn({right arrow over (k)})*E_(p)({right arrow over (k)}) represents a polarization density distribution, and is proportional to the macroscopic sum of the inhomogeneous electric dipole moments of the grating medium induced by the probe beam, E_(p)({right arrow over (k)}).

Typically, when the probe beam resembles one of the recording beams used for recording, the effect of the convolution is to reverse the cross correlation during recording, and the diffracted beam will substantially resemble the other recording beam used to record the hologram. When the probe beam has a different k-space distribution than the recording beams used for recording, the hologram may produce a diffracted beam that is substantially different than the beams used to record the hologram. Note also that while the recording beams are typically mutually coherent, the probe beam (and diffracted beam) is not so constrained. A multiwavelength probe beam may be analyzed as a superposition of single-wavelength beams, each obeying Equation (4) with a different k-sphere radius.

FIGS. 2A and 2B illustrate cases of Bragg-matched and Bragg-mismatched reconstructions, respectively, generated by illuminating the hologram depicted in FIGS. 1A and 1B. In both the Bragg-matched and Bragg-mismatched cases, the hologram is illuminated with a probe beam having a shorter wavelength than the recording beams used to record the hologram. The shorter wavelength corresponds to a longer wave vector. Accordingly, a probe k-sphere 172 has a greater radius than that of the recording k-sphere 170. Both the probe k-sphere 172 and the recording k-sphere 170 are indicated in FIGS. 2A and 2B.

FIG. 2A shows a case where the probe beam is designed to produce a diffracted beam k-space distribution 175 (represented as E_(d)({right arrow over (k)})) that is point-like and lies on the probe beam k-sphere 172. The diffracted beam k-space distribution 175 is produced according to the convolution of Equation (4). The probe beam has a k-space distribution 176 (represented as E_(p)({right arrow over (k)}) that is also point-like. In this case, the probe beam is said to be “Bragg-matched” to the hologram, and the hologram may produce significant diffraction, even though the probe beam wavelength differs from the wavelength of the recording beams used to record the hologram. As shown in FIG. 2A, the convolution operation may also be represented geometrically by the vector sum {right arrow over (k)}_(d)={right arrow over (k)}_(p)+{right arrow over (K)}_(G+), where {right arrow over (k)}_(d) represents a diffracted beam wave vector 177, {right arrow over (k)}_(p) represents a probe beam wave vector 178, and {right arrow over (K)}_(G+) represents a sideband grating vector 179.

FIG. 2A shows a k-space representation of a mirror-like diffraction (which can be referred to as a reflection) of the probe beam by the hologram, where the probe beam angle of incidence with respect to the k_(z) axis is equal to the diffracted beam angle of reflection with respect to the k_(z) axis. FIG. 2B shows a k-space representation of a Bragg-mismatched case, wherein a k-space polarization density distribution 180, which can be represented as Δn({right arrow over (k)})*E_(p)({right arrow over (k)}), does not lie on the probe k-sphere 172, and thus no significant diffraction of the probe beam occurs. This non-diffracted k-space distribution 180 in the Bragg-mismatched case illustrated in FIG. 2B is somewhat analogous to the diffracted beam k-space distribution 175 in the Bragg-matched case illustrated in FIG. 2A, but k-space distribution 180 should not be referred to as a diffracted beam k-space distribution because no significant diffraction of the probe beam occurs.

Comparing the Bragg-matched and Bragg-mismatched cases, it is evident that the hologram will only produce mirror-like diffraction over a very small range of input angles for a given probe wavelength, if at all. Those skilled in the art will recognize that this range may be somewhat extended by over-modulating the hologram, or by using a very thin recording layer; but that these steps may still not lead to mirror-like behavior over a larger range of wavelengths and angles. These steps may also lead to undesired chromatic dispersion.

Skew Mirror Embodiment in k-Space

FIGS. 1A, 1B, 2A, and 2B represent a reflection hologram constituted by a single sinusoidal grating. As illustrated, this hologram exhibits mirror-like reflectivity in a narrow band of wavelengths and incidence angles. The specific properties of such a hologram may be determined by application of the well-known coupled wave theory of Kogelnik. Conversely, embodiments of the present invention exhibit novel mirror-like reflectivity across relatively broad ranges of wavelengths and angles by creating a more complex grating structure comprising multiple gratings.

FIG. 3 shows a geometry illustrating the Bragg selectivity of a single sinusoidal grating. Grating medium 310 contains a single sinusoidal grating of thickness d which reflects incident light 324 of a single wavelength, λ₀, as principal reflected light 327. At the Bragg-matched condition, incident light 324 impinges at angle and θ_(i), reflects as reflected light 327 at angle θ_(r), both angles measured with respect to the z axis. Incident light 324 and reflected light 327 also define a reflective axis 338, about which the angular magnitudes of incidence θ_(i)′ and reflection θ_(r)′ are equal. Reflective axis 338 is thus an angular bisector of incident light 324 and reflected light 327.

As is known to those skilled in the art, the sinusoidal grating of FIG. 3 will exhibit both angular and wavelength Bragg selectivity. If incident light 324 impinges at non-Bragg-matched angle θ_(i)+Δθ_(i), the diffraction efficiency may be diminished compared to the Bragg-matched diffraction efficiency. The selectivity of a sinusoidal grating may be characterized by its angular Bragg selectivity, Δθ_(B), given by equation (5):

$\begin{matrix} {{\Delta\theta}_{B} = {\frac{{\lambda cos}\mspace{11mu}\theta_{r}}{n_{0}\; d\mspace{11mu}{\sin\left( {\theta_{i} - \theta_{r}} \right)}}.}} & (5) \end{matrix}$ Those skilled in the art will recognize that in a weakly-diffracting sinusoidal grating, the angle θ_(i)+Δθ_(B) represents the first null in the angular diffraction efficiency plot. The quantity Δθ_(B) can thus be said to represent the angular width of the sinusoidal grating in that diffraction can be greatly diminished when the angle of incidence deviates from the Bragg-matched angle θ_(i) by more than several times Δθ_(B). Similarly, for a weakly-diffracting sinusoidal grating, the skilled artisan would expect a reflective axis to vary considerably for monochromatic incident light whose angle of incidence varies by more than several times Δθ_(B).

Conversely, skew mirrors according to present disclosure exhibit relatively stable diffraction and substantially constant reflective axes for incident light whose angle of incidence varies by many times Δθ_(B). Some skew mirror embodiments exhibit substantially constant reflective axes across a range of incident light angles of incidence of 20×Δθ_(B). In embodiments, reflective axis angles across a range of incident light angles of incidence of 20×Δθ_(B) change by less than 0.250 degree; or by less than 0.10 degree; or by less than 0.025 degree.

Similarly, a sinusoidal grating may be characterized by its wavelength Bragg selectivity, Δλ_(B), given by equation (6):

$\begin{matrix} {{\Delta\lambda}_{B} = {\frac{\lambda_{0}^{2}\;\cos\mspace{11mu}\theta_{r}}{2n_{0}^{2}\; d\mspace{11mu}{\sin^{2}\left( {\theta_{i} - \theta_{r}} \right)}}.}} & (6) \end{matrix}$ Those skilled in the art will recognize that in a weakly-diffracting sinusoidal grating, the wavelength λ₀+Δθ_(B) represents the first null in the wavelength diffraction efficiency plot. The quantity Δθ_(B) can thus be said to represent the wavelength width of the sinusoidal grating in that no significant diffraction will occur when the incident wavelength deviates from the Bragg-matched wavelength λ₀ by more than several times Δθ_(B). Those skilled in the art will also recognize that equations (5) and (6) apply to changes in angle and wavelength only, respectively, and that changing both angle and wavelength simultaneously may result in another Bragg-matched condition.

A grating may also be characterized by its diffracted angle response. For a sinusoidal grating, the diffracted angle response may be expressed by equation (7): Δθ_(r) cos θ_(r)=−Δθ_(i) cos θ_(i).  (7) The diffracted angle response expresses the change in the angle of reflection, Δθ_(r), in response to small changes in the angle of incidence, Δθ_(i). In contrast, a true mirror has an angle response expressed by equation (8): Δθ_(r)=−Δθ_(i).  (8) A device that has a diffracted angle response substantially characterized by equation (7) may be said to exhibit grating-like reflective behavior, whereas a device that has a diffracted angle response substantially characterized by equation (8) may be said to exhibit mirror-like reflective behavior. A device exhibiting grating-like reflective behavior will necessarily also exhibit a reflective axis that changes with angle of incidence, unless that reflective axis is normal to the device surface, in which case cos θ_(r)=cos θ_(i). Accordingly, requirements for a relatively simple device that reflects light about a reflective axis not constrained to surface normal, and whose angle of reflection for angles of incidence spanning multiples of its angular Bragg selectivity is constant at wavelengths spanning multiples of its wavelength Bragg selectivity, may not be met by a single sinusoidal grating.

FIG. 3 illustrates a device geometry in a reflective configuration. Those skilled in the art will recognize that the preceding analysis also applies to device geometries in transmissive configurations and to device geometries in which one or both beams are guided by total internal reflection within the device.

FIGS. 4A and 4B illustrate operation of a skew mirror in k-space according to an embodiment. FIG. 4A shows two Δn({right arrow over (k)}) k-space distributions 488 for a hologram recorded in a grating medium and configured to produce multiwavelength mirror-like diffraction according to an embodiment. A red k-sphere 490, green k-sphere 492, and blue k-sphere 493 in FIGS. 4A and 4B indicate k-spheres corresponding to wavelengths of light residing in the red, green, and blue regions of the visible spectrum, respectively.

Instead of two Δn({right arrow over (k)}) k-space distributions constituting a single sinusoidal grating (and which therefore can be characterized as “point-like”), the Δn({right arrow over (k)}) k-space distributions 488 shown in FIG. 4A are situated along a substantially straight line in k-space, and thus can be characterized as “line segment-like”. In some embodiments, line segment-like Δn({right arrow over (k)}) k-space distributions comprise continuously-modulated sub-segments of a substantially straight line in k-space. In some embodiments, line segment-like Δn({right arrow over (k)}) k-space distributions substantially consist of point-like distributions situated along a substantially straight line in k-space. The line segment-like Δn({right arrow over (k)}) k-space distributions 488 are situated symmetrically about the origin, and thus may be realized as conjugate sidebands of a real-valued refractive index distribution in real space (represented as Δn({right arrow over (r)}). In some embodiments, the modulation may include absorptive and/or emissive components, and thus may not exhibit conjugate symmetry in k-space. The complex amplitude of the distribution may be uniform, or it may vary in amplitude and/or phase while still exhibiting substantially multiwavelength mirror-like diffraction according to embodiments of the present invention. In an embodiment, the line segment-like Δ_(n)({right arrow over (k)}) k-space distributions are situated substantially along the k_(z) axis, which, by convention, is the thickness direction of a grating medium.

FIG. 4B illustrates a multiwavelength mirror-like reflective property of the hologram. Illumination of the hologram by a collimated probe beam with point-like k-space distribution 476 (represented as E_(p)({right arrow over (k)})) results in a k-space polarization density distribution 480 (represented as Δn({right arrow over (k)})*E_(p)({right arrow over (k)})) according to Equation (4). Because the probe beam k-space distribution 476 is point-like, polarization density distribution 480 resembles a simple translation of Δn({right arrow over (k)}) k-space distribution 488 from the origin to the tip of probe beam wave vector 478 ({right arrow over (k)}_(p)). Then, also according to Equation (4), only the part of the k-space polarization density distribution 480 (Δn({right arrow over (k)})*E_(p)({right arrow over (k)})) intersecting the k-sphere 492 of the probe beam k-space distribution 476 (E_(p)({right arrow over (k)})) contributes to diffraction. This produces the diffracted beam k-space distribution 475, (E_(d)({right arrow over (k)}), constituting the diffracted beam. Because Δn({right arrow over (k)}) k-space distribution 488 resembles a line segment parallel to the k_(z) axis, it is evident that the magnitude of the angle of reflection 482 (θ_(r),) is substantially equal to the magnitude of the angle of incidence 481 (θ_(i),) so that the hologram exhibits mirror-like behavior. Furthermore, it is also evident that this property typically holds for any incidence angle and wavelength that produces any diffraction at all, and for any superposition of probe beams producing diffraction. A k-space polarization distribution Δn({right arrow over (k)})*E_(p)({right arrow over (k)}) will intersect the probe k-sphere at a single point with mirror-symmetry about the k_(x) axis (or about the k_(x), k_(y) plane in the 3D case). Thus, the hologram of FIG. 4A is configured to exhibit mirror-like behavior at a relatively broad range of wavelengths and angles, and thus constitutes a broadband holographic mirror.

Embodiments typically, but not necessarily, exhibit a gap in Δn ({right arrow over (k)}) k-space distribution 488 near the origin, as shown in FIG. 4A. The presence of the gap can limit performance at very high Δθ (i.e., grazing angles of both incidence and reflection).

According to an embodiment, a skew mirror Δn({right arrow over (k)}) k-space distribution may be rotated to an arbitrary angle with respect to the k_(x), k_(y), and k_(z) axes. In some embodiments, the Δn({right arrow over (k)}) k-space distribution is not perpendicular to the relevant reflecting surface in real space. In other words, the reflective axis of a skew mirror embodiment is not constrained to coincident with surface normal.

FIGS. 5A and 5B illustrate a skew mirror in k-space. FIGS. 5A and 5B are identical to FIGS. 4A and 4B, respectively, excepting that all distributions and vectors have been rotated by approximately 45° about the origin. Following the discussion of FIG. 4B, it is evident that the skew mirror of FIG. 5B also produces mirror-like diffraction for all probe beam wavelengths and angles that produce diffraction. The diffraction is mirror-like with respect to the reflective axis 461 defined by the line segment-like Δn({right arrow over (k)}) k-space distribution 488, i.e., the angle of incidence 481 magnitude with respect to the reflective axis 461 is equal to the angle of reflection 482 magnitude with respect to the reflective axis 461. FIG. 5B illustrates one such case.

FIG. 6A illustrates the operation of a skew mirror in real space. Skew mirror 610 is characterized by reflective axis 638 at angle −13° measured with respect to the z axis, which is normal to the skew mirror surface 612. Skew mirror 610 is illuminated with incident light 624 with internal incidence angle −26° measured with respect to the z axis. Principal reflected light 627 is reflected with internal reflection angle 180° measured with respect to the z axis.

FIG. 6B illustrates the skew mirror 610 of FIG. 6A in k-space. Line segment-like Δn({right arrow over (k)}) k-space distribution 688 passes through the origin, and has an angle of −13° with respect to the z axis, equal to that of reflective axis 638. Recording k-sphere 670 is the k-sphere corresponding to the writing wavelength of 405 nm. A red k-sphere 690, green k-sphere 692, and blue k-sphere 693 in FIGS. 6B and 6D indicate k-spheres corresponding to wavelengths of light residing in the red, green, and blue regions of the visible spectrum, respectively.

FIG. 6C illustrates a highly magnified portion of FIG. 6B showing the left intersection between recording k-sphere 670 and line segment-like Δn({right arrow over (k)}) k-space distribution 688 according to an embodiment. In this view, line segment-like Δn({right arrow over (k)}) k-space distribution 688 can be seen to be include multiple discrete holograms. Each of the multiple discreet holograms 605 is represented by a horizontal line demarking the first null-to-first null spacing of the hologram in the k_(z) direction. In some embodiments, the spacing of the discrete holograms may be higher or lower than illustrated in 6C. In some embodiments, the spacing may be low enough to create gaps in line segment-like Δn({right arrow over (k)}) k-space distribution 688. In some embodiments with gaps, the use of broadband illumination may substantially mask any effect of the gaps upon the reflected light. In some embodiments, this approach may result in a net diffraction efficiency increase. In other embodiments, the spacing of the discrete holograms may be so dense as to approximate or be equivalent to a continuous distribution.

FIG. 6D illustrates the reflection of blue incident light by the skew mirror of FIG. 6A in k-space. Incident light having a probe beam wave vector 678 impinges with an internal incidence angle of −26° measured with respect to the z axis. The tip of probe beam wave vector 678 lies on blue k-sphere 693, indicating the position of point-like probe beam k-space distribution 676 (E_(p)({right arrow over (k)})). Polarization density distribution 680 is given by the convolution Δn({right arrow over (k)})*E_(p) ({right arrow over (k)}) which resembles line segment-like Δn({right arrow over (k)}) k-space distribution 688 (seen in FIG. 6C) translated to the tip of probe beam wave vector 678. Principal reflected light having diffracted beam wave vector 677 is determined from equation (4) by evaluating polarization density distribution 680 at blue k-sphere 693. Principal reflected light having diffracted beam wave vector 677 is reflected with internal propagation angle 180° measured with respect to the z axis.

Persons skilled in the art will recognize that the term probe beam, typically used here when describing skew mirror properties in k-space, is analogous to the term incident light, which is typically used here when describing skew mirror reflective properties in real space. Similarly, the term diffracted beam, typically used here when describing skew mirror properties in k-space, is analogous to the term principal reflected light, typically used here when describing skew mirror properties in real space. Thus when describing reflective properties of a skew mirror in real space, it is typical to state that incident light is reflected by a hologram (or other grating structure) as principal reflected light, though to state that a probe beam is diffracted by the hologram to produce a diffracted beam says essentially the same thing. Similarly, when describing reflective properties of a skew mirror in k-space, it is typical to state that a probe beam is diffracted by a hologram (or other grating structure) to produce a diffracted beam, though to state that incident light is reflected by the grating structure to produce principal reflected light has the same meaning in the context of embodiments of the present invention.

As shown in FIG. 6D, probe beam wave vector 678 and diffracted beam wave vector 677 necessarily form the legs of a substantially isosceles triangle with line segment-like polarization density distribution 680 as the base. The equal angles of this triangle are necessarily congruent with the angle of incidence, 608, and angle of reflection 609, both measured with respect to reflective axis 638. Thus, skew mirror 610 reflects light in a substantially mirror-like manner about reflective axis 638.

The isosceles triangle construction of FIG. 6D obtains whenever Δn({right arrow over (k)}) k-space distribution 688 substantially resembles a segment of a line passing through the origin, as shown in FIG. 6C. Polarization density distribution 680 hence substantially resembles the straight base of an isosceles triangle, leading to mirror-like reflection about reflective axis 638 for any incident internal wave vectors of any length that diffracts. In some embodiments, dispersion of the grating medium may cause internal wave vectors of the same direction but differing lengths to refract in different directions in an external medium according to Snell's law. Similarly, dispersion may cause external wave vectors of the same direction and differing lengths to refract in different directions in the internal grating medium. Accordingly, if it is desired to minimize the effects of dispersion in a skew mirror, it may be desirable to impart a curve to line segment-like Δn ({right arrow over (k)}) k-space distribution 688, or to otherwise deviate from a line that passes through the origin. Such an approach may reduce net angular dispersion in reflections involving external refraction according to some metric. Since the dispersion of useful grating media is typically quite low, the deviation from a straight line passing through the origin may be small.

FIG. 7A illustrates the reflection of green incident light by the skew mirror of FIG. 6A in k-space. Incident light with wave vector 778A impinges with internal propagation angle −35° measured with respect to the z axis. Principal reflected light with wave vector 777A is reflected with internal propagation angle −171° measured with respect to the z axis. The magnitudes of angle of incidence 708A and angle of reflection 709A are both substantially equal to 22 degrees measured with respect to reflective axis 638, thus constituting a mirror-like reflection about reflective axis 638. Polarization density distribution 780A is also illustrated in FIG. 7A.

FIG. 7B illustrates the reflection of red incident light by the skew mirror of FIG. 6A in k-space. Incident light having probe beam wave vector 778B impinges with internal propagation angle −35° measured with respect to the z axis. Principal reflected light with wave vector 777A is reflected with internal propagation angle −171° measured with respect to the z axis. The magnitudes of angle of incidence 708A and angle of reflection 709A are both substantially equal to 22 degrees measured with respect to reflective axis 638, thus constituting a mirror-like reflection about reflective axis 638. Polarization density distribution 780A is also illustrated in FIG. 7A.

FIGS. 7A and 7B show the reflection of green and red light at the same angles of incidence and reflection, illustrating the achromatic reflection property of the skew mirror. Those skilled in the art will recognize that the geometrical constructions of FIGS. 6A-D and 7A-B will produce mirror-like reflection at all angle/wavelength combinations that produce reflection, including angles and wavelengths not specifically illustrated.

Skew Mirror Optical Properties

Embodiments of a skew mirror effect a mirror-like reflection with respect to internal propagation angles, external angles must be determined using Snell's law at the relevant boundaries. Because of this, a skew mirror may introduce aberrations, dispersion, and/or field distortion to external wavefronts. In some embodiments, aberrations, dispersion, and/or field distortions may be mitigated by the use of compensating optics. In some embodiments, the compensating optics may include another skew mirror in a symmetric relationship.

A relatively thin skew mirror may introduce lowered angular resolution in the reflected beam in proportion to the beam's projection onto the thin axis. In some cases it may be advantageous to increase the thickness of the recording layer in order to mitigate this effect.

Skew Mirror Reflectivity

Embodiments of a skew mirror may be either fully or partially reflective. Embodiments of a skew mirror may require relatively high dynamic range recording medium to achieve high reflectivity over a relatively wide wavelength bandwidth and angle range. In an embodiment, a skew mirror with an angular range spanning 105° at 405 nm down to 20° at 650 nm may require 183 individual holograms in a 200 μm recording layer. This configuration has a reflectivity of approximately 7.5% using a state-of-the-art photosensitive recording medium with a maximum refractive index modulation of 0.03. In some embodiments, increasing recording medium thickness may not lead to increased reflectivity since diffractive selectivity also increases with thickness.

Skew Mirror Applications

The preceding exposition pertains to internal wavelengths and propagation angles, although in one case a slab-like hologram with thickness in the z direction was described. Many other configurations are possible within the scope of the invention. Without implying limitation, a few exemplary embodiments are illustrated here.

FIG. 8A illustrates an embodiment referred to as a skew window comprising grating structure 805 in a grating medium, and including a reflective axis 861 about which incident light is symmetrically refracted. The skew window is a transmissive analog of the skew mirror. FIG. 8B shows a skew coupler embodiment, which uses a skew mirror to couple external light into or out of a waveguide 894. Transmissive skew couplers are also possible. FIG. 8C shows a skew prism embodiment, which may fold an optical path and/or invert an image.

FIG. 9A illustrates a pupil relay embodiment formed by a slab waveguide 994 with two skew couplers, each of which comprises a grating medium 910 having a reflective axis 961 that differs from surface normal of the grating medium. Since this device is configured to relay input rays to output rays with a uniform 1:1 mapping, it can transmit an image at infinity through the waveguide 994 to the eye or other sensor. Such a configuration may be useful for head mounted displays (HMDs), among other applications. In the reverse direction, it may relay an image of the eye, possibly for the purposes of eye tracking. FIG. 9B shows a skew mirror 900 used as a concentrator/diffuser, which can transform a large dim beam into a bright small one, and/or vice-versa.

FIGS. 10A and 10B illustrate an angle filter embodiment of a skew mirror. In FIG. 10A, a Δn({right arrow over (k)}) k-space 1088 distribution is indicated with a higher low frequency cut-off (i.e., larger center gap) compared to the distribution illustrated in FIG. 8A. As a consequence, the skew mirror will reflect only the low θ (i.e., near normal incidence) angular components of narrow band incident beam E_(inc), into reflected beam E_(r), while transmitting high θ angular components in E_(t). One skilled in the art will readily discern that an arbitrary circularly-symmetric transfer function may be so realized by modulating the amplitude and/or phase of the line segment-like Δ_(n)({right arrow over (k)}) distribution according to an embodiment of the invention. Angular filtering may also be accomplished with skew mirrors, and in configurations involving multiple skew mirrors recorded in one or more media. These configurations may not be constrained to be circularly-symmetric, and may achieve some level of achromatic operation.

A First Embodiment Skew Mirror

Inventive aspects of a first embodiment skew mirror include the mirror being configured to reflect incident light having a wavelength of 532 nm and incident light having a wavelength of 513 nm about reflective axes that collectively have a mean reflective axis angle of +13.73 degrees relative to surface normal. In a further inventive aspect, the mean reflective axis angle (+13.759 degrees) for 532 nm light incident upon the skew mirror at internal angles of incidence ranging from −4.660 to +1.933 degrees differs by only 0.066 degree from the mean reflective axis angle (+13.693 degrees) for 513 nm light incident upon the skew mirror at the same angles of incidence as the 532 nm incident light. The reflective axes are thus substantially constant for the 532 nm to 513 nm wavelength range, a condition that obtains for internal angles of incidence (relative to surface normal) from −4.660 degrees to +1.993 degrees.

The first embodiment skew mirror 1100 is illustrated in FIGS. 11A and 11B. The first embodiment skew mirror 1100 comprises a grating structure 1105 (shown by diagonal hatch lines in FIGS. 11A and 11B) residing in a grating medium 1110. For purposes of clarity, the diagonal hatch lines are omitted in a region within the grating medium 1110 proximate figure elements indicating light, axes, and angles. However, persons skilled in the art will recognize that the grating structure 1105 typically occupies the region described above. The grating structure 1105 of the first embodiment includes multiple holograms that at least partially spatially overlap with each other in the grating medium 1110.

The multiple holograms are recorded into the grating medium internal volume and thus extend below the grating medium surface 1112. Accordingly, they are sometimes referred to as volume holograms. The multiple holograms of the first embodiment comprise forty eight (48) volume holograms, recorded with recording beams having a wavelength of 405 nm. Each of the 48 volume holograms typically at least partially spatially overlaps all others of the 48 volume holograms in the grating medium 1110. In some embodiments, each of the multiple holograms at least partially spatially overlaps at least one, but not all, of the other of the multiple holograms. Recording the 48 holograms of the first embodiment skew mirror is described below in a first method of making a skew mirror. In some embodiments, the grating structure includes between 1 and 48 holograms; or between 4 and 25 holograms; or at least 5 holograms; or at least 9 holograms; or at least 11 holograms; or at least 24 holograms.

The first embodiment grating medium 1110 is a proprietary photosensitive polymeric optical recording medium, designated AK174-200, available from Akonia Holographics, LLC (Longmont, Colo.). The AK174-200 recording medium of the first embodiment is approximately 200 μm thick, has an M/# of approximately 18, and a refractive index of approximately 1.50 for 405 nm light. Optical recording mediums such as the AK174-200 medium are a type of grating medium in which grating structures can be recorded by optical means. Grating mediums are typically, but not necessarily, at least 70 μm thick to approximately 1.2 mm thick. The AK174-200 medium typically undergoes relatively little shrinkage (usually about 0.1% to 0.2%) as a result of recording volume holograms. Variations of grating mediums include, but are not limited to, photorefractive crystals, dichromated gelatin, photo-thermo-refractive glass, and film containing dispersed silver halide particles.

Variations of the first embodiment skew mirror 1100 may include an additional layer such as a glass cover or glass substrate (not shown in FIGS. 11A and 11B). The additional layer may serve to protect the grating medium from contamination, moisture, oxygen, reactive chemical species, damage, and the like. The additional layer is typically refractive index matched to the grating medium 1110. Because the refractive index for the additional layer is usually very close to the refractive index of the grating medium, refraction of light at the interface of the additional layer and the grating medium can sometimes be ignored. For the first embodiment, refractive indices for both the additional layer and the grating medium are approximately 1.5 for light having a wavelength of 405 nm. For clarity, the additional layer is not shown in FIGS. 11A and 11B.

As shown in FIG. 11A, the grating structure 1105 of the first embodiment has the physical property of being configured to reflect a first incident light 1124A, 1124B, about a first reflective axis 1138 (shown in broken line). The first incident light has a first wavelength of 532 nm and is incident upon the grating medium 1110 at a specific site 1117. The first reflective axis 1138 differs from surface normal 1122 of the grating medium by a first reflective axis angle 1135 of +13.759 degrees (internal, relative to surface normal), where the first incident light has an first internal angle of incidence 1125A, 1125B relative to surface normal, from −4.660 degrees (shown as first incident light 1124A) to +1.933 degrees (shown as first incident light 1124B), resulting in a range of 6.593 degrees. The first internal angles of incidence for the first incident light include one hundred (100) different internal angles spaced at angle intervals of about 0.067 degrees, from −4.660 degrees to +1.933 degrees, as shown in Table 1. In some variations of the first embodiment skew mirror, the first internal angles of incidence for the first incident light include ten (10) different internal angles spaced at angle intervals of about 0.67 degrees, from −4.660 degrees to +1.933 degrees. Throughout this specification and appended claims, identified angles and angle values refer to internal angles relative to surface normal, unless clearly indicated otherwise.

As shown FIG. 11A, first incident light 1124A, having a first internal angle of incidence of 1125A of −4.660 degrees relative to surface normal, is reflected by the grating structure 1105 as first reflected light 1127A, having a first internal angle of reflection 1126A of +32.267 degrees relative to surface normal. First incident light 1124B, having a first internal angle of incidence 1125B relative to surface normal of +1.933 degrees, is reflected as first reflected light 1127B having a first internal angle of reflection 1126B of +25.668 degrees. First reflected light 1127A, 1127B has the first wavelength, i.e. in the first embodiment the first reflected light has a wavelength of 532 nm. First incident light angles, first reflected light angles, and first reflective axis angles for the first embodiment skew mirror are shown in Table 1.

TABLE 1 ANGLES OF FIRST INCIDENT LIGHT, FIRST REFLECTED LIGHT, AND FIRST REFLECTIVE AXIS, FOR A FIRST EMBODIMENT SKEW MIRROR; WAVELENGTH = 532 nm; AK174-200 RECORDING MEDIUM; N = 100 Angle of Angle Of Angle Of Angle of Reflection First First Incidence Reflection Incidence of First Internal First Internal of First of First of First Reflected First Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 25.668 13.800 1.933 −11.867 11.867 2.900 40.521 21.711 25.680 13.773 1.866 −11.907 11.907 2.800 40.542 21.671 25.691 13.746 1.800 −11.946 11.946 2.701 40.563 21.632 25.814 13.774 1.733 −12.041 12.041 2.600 40.782 21.691 25.938 13.803 1.667 −12.136 12.136 2.501 41.003 21.752 26.005 13.802 1.600 −12.202 12.202 2.400 41.122 21.761 25.904 13.719 1.533 −12.185 12.185 2.300 40.942 21.621 25.971 13.719 1.466 −12.252 12.252 2.200 41.062 21.631 26.094 13.747 1.400 −12.347 12.347 2.101 41.283 21.692 26.216 13.775 1.333 −12.442 12.442 2.000 41.502 21.751 26.339 13.803 1.267 −12.536 12.536 1.901 41.723 21.812 26.350 13.775 1.200 −12.575 12.575 1.800 41.742 21.771 26.472 13.803 1.134 −12.669 12.669 1.701 41.963 21.832 26.538 13.802 1.067 −12.736 12.736 1.600 42.082 21.841 26.660 13.830 1.001 −12.830 12.830 1.501 42.303 21.902 26.780 13.857 0.933 −12.924 12.924 1.399 42.521 21.960 26.738 13.802 0.867 −12.935 12.935 1.301 42.443 21.872 26.803 13.801 0.800 −13.001 13.001 1.200 42.561 21.881 26.923 13.829 0.734 −13.095 13.095 1.101 42.781 21.941 26.989 13.828 0.667 −13.161 13.161 1.000 42.901 21.951 26.946 13.773 0.601 −13.173 13.173 0.901 42.822 21.862 27.066 13.800 0.533 −13.266 13.266 0.800 43.041 21.921 26.913 13.690 0.467 −13.223 13.223 0.701 42.762 21.732 27.088 13.744 0.400 −13.344 13.344 0.600 43.081 21.841 27.263 13.798 0.334 −13.464 13.464 0.501 43.402 21.952 27.436 13.852 0.267 −13.585 13.585 0.400 43.721 22.061 27.230 13.715 0.201 −13.515 13.515 0.301 43.342 21.822 27.241 13.687 0.133 −13.554 13.554 0.200 43.361 21.781 27.416 13.742 0.067 −13.674 13.674 0.101 43.683 21.892 27.589 13.794 0.000 −13.794 13.794 0.000 44.002 22.001 27.600 13.766 −0.067 −13.833 13.833 −0.100 44.022 21.961 27.664 13.766 −0.133 −13.899 13.899 −0.200 44.142 21.971 27.837 13.818 −0.200 −14.018 14.018 −0.300 44.462 22.081 27.955 13.844 −0.267 −14.111 14.111 −0.400 44.682 22.141 28.074 13.870 −0.333 −14.203 14.203 −0.499 44.903 22.202 28.030 13.815 −0.401 −14.215 14.215 −0.601 44.822 22.111 28.042 13.788 −0.467 −14.254 14.254 −0.700 44.844 22.072 28.106 13.786 −0.533 −14.320 14.320 −0.800 44.964 22.082 28.224 13.812 −0.600 −14.412 14.412 −0.900 45.184 22.142 28.288 13.811 −0.667 −14.477 14.477 −1.000 45.304 22.152 28.298 13.783 −0.733 −14.516 14.516 −1.100 45.324 22.112 28.362 13.781 −0.800 −14.581 14.581 −1.200 45.444 22.122 28.427 13.781 −0.866 −14.646 14.646 −1.299 45.566 22.134 28.437 13.752 −0.933 −14.685 14.685 −1.400 45.585 22.093 28.607 13.804 −0.999 −14.803 14.803 −1.499 45.906 22.204 28.670 13.802 −1.067 −14.868 14.868 −1.600 46.026 22.213 28.734 13.800 −1.133 −14.933 14.933 −1.700 46.146 22.223 28.797 13.798 −1.200 −14.998 14.998 −1.800 46.266 22.233 28.808 13.771 −1.266 −15.037 15.037 −1.899 46.287 22.194 28.923 13.795 −1.333 −15.128 15.128 −2.000 46.506 22.253 28.829 13.715 −1.399 −15.114 15.114 −2.099 46.327 22.114 28.996 13.765 −1.466 −15.231 15.231 −2.200 46.646 22.223 29.007 13.737 −1.532 −15.270 15.270 −2.299 46.667 22.184 29.069 13.735 −1.600 −15.335 15.335 −2.400 46.786 22.193 29.028 13.681 −1.666 −15.347 15.347 −2.499 46.707 22.104 29.142 13.705 −1.733 −15.438 15.438 −2.600 46.926 22.163 29.309 13.755 −1.799 −15.554 15.554 −2.699 47.247 22.274 29.475 13.804 −1.866 −15.670 15.670 −2.800 47.566 22.383 29.330 13.699 −1.932 −15.631 15.631 −2.899 47.287 22.194 29.392 13.696 −1.999 −15.696 15.696 −3.000 47.406 22.203 29.558 13.746 −2.065 −15.812 15.812 −3.099 47.727 22.314 29.670 13.769 −2.133 −15.902 15.902 −3.200 47.946 22.373 29.630 13.716 −2.199 −15.914 15.914 −3.299 47.867 22.284 29.640 13.687 −2.266 −15.953 15.953 −3.400 47.886 22.243 29.752 13.710 −2.333 −16.043 16.043 −3.500 48.106 22.303 29.916 13.759 −2.399 −16.158 16.158 −3.600 48.426 22.413 29.825 13.680 −2.465 −16.145 16.145 −3.699 48.247 22.274 29.988 13.728 −2.532 −16.260 16.260 −3.800 48.566 22.383 30.151 13.776 −2.598 −16.374 16.374 −3.899 48.887 22.494 30.160 13.747 −2.665 −16.413 16.413 −4.000 48.906 22.453 30.170 13.719 −2.732 −16.451 16.451 −4.100 48.926 22.413 30.332 13.767 −2.799 −16.565 16.565 −4.200 49.246 22.523 30.394 13.765 −2.865 −16.629 16.629 −4.299 49.368 22.535 30.302 13.685 −2.932 −16.617 16.617 −4.400 49.187 22.394 30.363 13.683 −2.998 −16.681 16.681 −4.499 49.308 22.405 30.474 13.704 −3.065 −16.769 16.769 −4.600 49.527 22.464 30.634 13.752 −3.131 −16.883 16.883 −4.699 49.848 22.575 30.694 13.748 −3.198 −16.946 16.946 −4.800 49.967 22.584 30.654 13.695 −3.264 −16.959 16.959 −4.899 49.888 22.495 30.814 13.741 −3.331 −17.072 17.072 −5.000 50.208 22.604 30.874 13.738 −3.397 −17.135 17.135 −5.099 50.329 22.615 30.834 13.685 −3.464 −17.149 17.149 −5.200 50.248 22.524 30.894 13.682 −3.530 −17.212 17.212 −5.299 50.369 22.535 31.051 13.727 −3.597 −17.324 17.324 −5.400 50.688 22.644 31.160 13.749 −3.663 −17.411 17.411 −5.499 50.909 22.705 31.169 13.720 −3.730 −17.450 17.450 −5.600 50.928 22.664 31.180 13.692 −3.796 −17.488 17.488 −5.699 50.949 22.625 31.336 13.736 −3.863 −17.599 17.599 −5.800 51.268 22.734 31.443 13.757 −3.929 −17.686 17.686 −5.899 51.488 22.795 31.549 13.777 −3.996 −17.772 17.772 −6.000 51.706 22.853 31.704 13.821 −4.062 −17.883 17.883 −6.099 52.027 22.964 31.713 13.792 −4.129 −17.921 17.921 −6.200 52.046 22.923 31.723 13.764 −4.195 −17.959 17.959 −6.299 52.067 22.884 31.636 13.687 −4.262 −17.949 17.949 −6.400 51.886 22.743 31.695 13.684 −4.327 −18.011 18.011 −6.499 52.007 22.754 31.848 13.727 −4.395 −18.121 18.121 −6.600 52.326 22.863 31.858 13.699 −4.460 −18.159 18.159 −6.699 52.347 22.824 31.963 13.718 −4.527 −18.245 18.245 −6.800 52.566 22.883 32.116 13.762 −4.593 −18.355 18.355 −6.899 52.888 22.995 32.267 13.804 −4.660 −18.464 18.464 −7.000 53.207 23.104 Mean = 13.759 Mean = 22.234 Std. 0.047 Dev. =

Incident light and its reflection are bisected by the reflective axis such that the internal angle of incidence of the incident light relative to the reflective axis has the same magnitude as the internal angle of reflection of the reflected light relative to the reflective axis. Thus it can be said that the incident light and its reflection exhibit bilateral symmetry about the reflective axis.

As shown in FIG. 11B, the grating structure 1105 of the first embodiment is further configured to reflect second incident light 1130A, 1130B about a second reflective axis 1139. The second incident light has a second wavelength of 513 nm and is incident upon the grating medium 1110 at the specific site 1117. The specific site 1117 includes an area of the grating medium surface 1112 upon which both the first and second incident light shine. The second reflective axis 1139 differs from surface normal 1122 of the grating medium by a second reflective axis angle 1136 of +13.693 degrees (internal) relative to surface normal, where the second incident light has a second internal angle of incidence, relative to surface normal, from −4.660 degrees to +1.933 degrees. The second internal angle of incidence includes one hundred (100) different internal angles spaced at angle intervals of approximately 0.067 degrees, from −4.660 degrees to +1.933 degrees. In some variations of the first embodiment skew mirror, the second internal angles of incidence for the second incident light include ten (10) different internal angles spaced at angle intervals of about 0.67 degrees, from −4.660 degrees to +1.933 degrees.

As shown in FIG. 11B, second incident light 1130A, having a second internal angle of incidence 1128A of −4.660 degrees relative to surface normal, is reflected by the grating structure 1105 as second reflected light 1133A, having a second internal angle of reflection 1133A of +32.075 degrees relative to surface normal. Second incident light 1130B, having a second internal angle of incidence 1128B relative to surface normal of +1.933 degrees, is reflected as second reflected light 1133B having a second internal angle of reflection 1129B of +25.273 degrees. Second reflected light 1133A, 1133B has the second wavelength, i.e. in the first embodiment the second reflected light has a wavelength of 513 nm. Second incident light angles, second reflected light angles, and second reflective axis angles for the first embodiment skew mirror, are shown in Table 2.

TABLE 2 ANGLES OF SECOND INCIDENT LIGHT, SECOND REFLECTED LIGHT, AND SECOND REFLECTIVE AXIS, FOR A FIRST EMBODIMENT SKEW MIRROR; WAVELENGTH = 513 nm; AK174-200 RECORDING MEDIUM; N = 100 Angle of Angle Of Angle Of Angle of Reflection Second Second Incidence Reflection Incidence of Second Internal Second Internal of Second of Second of Second Reflected Second Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 25.273 13.603 1.933 −11.670 11.670 2.900 39.821 21.361 25.341 13.604 1.866 −11.737 11.737 2.800 39.942 21.371 25.466 13.633 1.800 −11.833 11.833 2.701 40.163 21.432 25.645 13.689 1.733 −11.956 11.956 2.600 40.481 21.541 25.769 13.718 1.667 −12.051 12.051 2.501 40.702 21.602 25.780 13.690 1.600 −12.090 12.090 2.400 40.721 21.561 25.959 13.746 1.533 −12.213 12.213 2.300 41.041 21.671 25.915 13.691 1.466 −12.224 12.224 2.200 40.961 21.581 25.982 13.691 1.400 −12.291 12.291 2.100 41.081 21.591 26.160 13.746 1.333 −12.413 12.413 2.000 41.400 21.700 26.171 13.719 1.267 −12.452 12.452 1.900 41.420 21.660 26.181 13.691 1.200 −12.491 12.491 1.800 41.439 21.620 26.249 13.691 1.134 −12.557 12.557 1.701 41.560 21.631 26.259 13.663 1.067 −12.596 12.596 1.600 41.579 21.590 26.438 13.719 1.001 −12.718 12.718 1.501 41.900 21.701 26.448 13.691 0.933 −12.757 12.757 1.400 41.919 21.660 26.515 13.691 0.867 −12.824 12.824 1.301 42.040 21.671 26.636 13.718 0.800 −12.918 12.918 1.200 42.259 21.730 26.592 13.663 0.734 −12.929 12.929 1.101 42.180 21.641 26.769 13.718 0.667 −13.051 13.051 1.000 42.500 21.750 26.780 13.690 0.601 −13.090 13.090 0.901 42.520 21.711 26.845 13.689 0.533 −13.156 13.156 0.800 42.639 21.720 26.912 13.690 0.467 −13.222 13.222 0.701 42.760 21.731 26.977 13.689 0.400 −13.289 13.289 0.600 42.879 21.740 26.989 13.661 0.334 −13.327 13.327 0.501 42.900 21.701 27.108 13.687 0.266 −13.421 13.421 0.399 43.118 21.759 27.229 13.715 0.201 −13.514 13.514 0.301 43.340 21.821 27.240 13.686 0.133 −13.553 13.553 0.200 43.359 21.780 27.360 13.714 0.067 −13.646 13.646 0.101 43.580 21.841 27.425 13.713 0.000 −13.713 13.713 0.000 43.700 21.850 27.490 13.712 −0.066 −13.778 13.778 −0.099 43.820 21.861 27.555 13.711 −0.133 −13.844 13.844 −0.200 43.939 21.870 27.565 13.683 −0.200 −13.883 13.883 −0.300 43.959 21.830 27.630 13.682 −0.267 −13.949 13.949 −0.400 44.079 21.840 27.750 13.709 −0.333 −14.041 14.041 −0.499 44.300 21.901 27.760 13.680 −0.400 −14.080 14.080 −0.600 44.319 21.860 27.825 13.680 −0.466 −14.146 14.146 −0.699 44.440 21.871 27.889 13.678 −0.533 −14.211 14.211 −0.800 44.559 21.880 28.007 13.703 −0.600 −14.303 14.303 −0.900 44.778 21.939 28.017 13.675 −0.667 −14.342 14.342 −1.000 44.798 21.899 28.135 13.701 −0.733 −14.434 14.434 −1.100 45.018 21.959 28.253 13.726 −0.800 −14.526 14.526 −1.200 45.238 22.019 28.264 13.699 −0.866 −14.565 14.565 −1.299 45.259 21.980 28.274 13.670 −0.933 −14.604 14.604 −1.400 45.278 21.939 28.338 13.669 −0.999 −14.669 14.669 −1.499 45.399 21.950 28.455 13.694 −1.067 −14.761 14.761 −1.600 45.619 22.010 28.572 13.719 −1.133 −14.852 14.852 −1.700 45.839 22.070 28.635 13.718 −1.200 −14.917 14.917 −1.800 45.959 22.080 28.646 13.690 −1.267 −14.956 14.956 −1.900 45.979 22.040 28.709 13.688 −1.333 −15.021 15.021 −2.000 46.099 22.050 28.720 13.660 −1.399 −15.060 15.060 −2.099 46.120 22.011 28.835 13.684 −1.466 −15.151 15.151 −2.200 46.339 22.070 28.899 13.683 −1.532 −15.216 15.216 −2.299 46.460 22.081 29.013 13.707 −1.600 −15.307 15.307 −2.400 46.679 22.140 29.024 13.679 −1.666 −15.345 15.345 −2.499 46.700 22.101 29.087 13.677 −1.733 −15.410 15.410 −2.600 46.819 22.110 29.150 13.675 −1.799 −15.474 15.474 −2.699 46.940 22.121 29.264 13.699 −1.866 −15.565 15.565 −2.800 47.159 22.180 29.326 13.697 −1.932 −15.629 15.629 −2.899 47.280 22.191 29.388 13.694 −1.999 −15.694 15.694 −3.000 47.399 22.200 29.502 13.718 −2.065 −15.784 15.784 −3.099 47.620 22.261 29.667 13.767 −2.133 −15.900 15.900 −3.200 47.939 22.370 29.678 13.739 −2.199 −15.938 15.938 −3.299 47.960 22.331 29.790 13.762 −2.266 −16.028 16.028 −3.400 48.180 22.390 29.647 13.657 −2.333 −15.990 15.990 −3.500 47.900 22.200 29.760 13.680 −2.399 −16.079 16.079 −3.600 48.120 22.260 29.822 13.678 −2.465 −16.143 16.143 −3.699 48.241 22.271 29.882 13.675 −2.532 −16.207 16.207 −3.800 48.360 22.280 29.944 13.672 −2.599 −16.271 16.271 −3.900 48.480 22.290 30.056 13.695 −2.665 −16.361 16.361 −4.000 48.700 22.350 30.066 13.667 −2.732 −16.399 16.399 −4.100 48.721 22.311 30.229 13.715 −2.799 −16.514 16.514 −4.200 49.041 22.421 30.290 13.713 −2.865 −16.577 16.577 −4.299 49.162 22.432 30.349 13.709 −2.932 −16.641 16.641 −4.400 49.280 22.440 30.360 13.681 −2.998 −16.679 16.679 −4.499 49.301 22.401 30.420 13.677 −3.065 −16.742 16.742 −4.600 49.420 22.410 30.531 13.700 −3.131 −16.831 16.831 −4.699 49.641 22.471 30.590 13.696 −3.198 −16.894 16.894 −4.800 49.760 22.480 30.651 13.694 −3.264 −16.957 16.957 −4.899 49.881 22.491 30.710 13.690 −3.331 −17.021 17.021 −5.000 50.000 22.500 30.820 13.712 −3.397 −17.109 17.109 −5.099 50.221 22.561 30.830 13.683 −3.464 −17.147 17.147 −5.200 50.240 22.520 30.939 13.705 −3.530 −17.235 17.235 −5.299 50.461 22.581 30.949 13.676 −3.597 −17.273 17.273 −5.400 50.480 22.540 31.009 13.673 −3.663 −17.336 17.336 −5.499 50.602 22.552 31.068 13.669 −3.730 −17.399 17.399 −5.600 50.721 22.561 31.225 13.714 −3.797 −17.511 17.511 −5.700 51.041 22.671 31.284 13.710 −3.863 −17.573 17.573 −5.800 51.161 22.681 31.293 13.682 −3.929 −17.611 17.611 −5.900 51.181 22.641 31.352 13.678 −3.996 −17.674 17.674 −6.000 51.302 22.651 31.460 13.699 −4.062 −17.761 17.761 −6.099 51.522 22.712 31.517 13.694 −4.129 −17.823 17.823 −6.200 51.641 22.721 31.528 13.667 −4.195 −17.861 17.861 −6.299 51.662 22.682 31.682 13.710 −4.262 −17.972 17.972 −6.400 51.981 22.791 31.692 13.682 −4.327 −18.010 18.010 −6.499 52.002 22.752 31.798 13.701 −4.395 −18.096 18.096 −6.600 52.221 22.811 31.904 13.722 −4.460 −18.182 18.182 −6.699 52.442 22.872 31.913 13.693 −4.527 −18.220 18.220 −6.800 52.461 22.831 31.970 13.689 −4.593 −18.282 18.282 −6.899 52.582 22.842 32.075 13.707 −4.660 −18.368 18.368 −7.000 52.801 22.901 Mean = 13.693 Mean = 22.110 Std. 0.025 Dev. =

The first wavelength (λ₁=532 nm) differs from the second wavelength (λ₂=513 nm) by 19 nm, which can be represented by a value referred to as a wave fraction (WF), defined as WF=|λ₁−λ₂|/[(λ₁+λ₂)/2]. Thus where the multiple wavelengths include a first wavelength of 532 nm and a second wavelength of 513 nm, WF=0.036. Similarly, where the multiple wavelengths consist of a continuous spectrum from 390 nm or less to at least 700 nm, WF≥0.57. Embodiments include, but are not limited to, variations in which WF≥0.005; WF≥0.010; WF≥0.030; WF≥0.10; WF≥0.250; WF≥1.0; or WF≥2.0. The wave fraction (WF) defined by a first (λ₁) and a second (λ₂) wavelength in the range may, but does not necessarily, includes a continuous spectrum of wavelengths between λ₁ and λ₂.

The second reflective axis angle 1136 differs from the first reflective axis angle 1135 by 0.066 degree. Accordingly, the second reflective axis is substantially coincident with the first reflective axis, meaning that the second reflective axis angle 1136 differs from first reflective axis angle 1135 by 1.0 degree or less. Such small difference between reflecting axis angles across a range of wavelengths (in this case, across a WF of 0.039) means that the grating structure acts as a nondispersive mirror. For some applications, the difference between reflective axis angles should be 0.250 degree or less for WF=0.030. Similarly, for some other applications, the difference between reflective axis angles should equal 0.10 degree or less for WF=0.030.

Relative to the first reflective axis, internal angles of incidence of the first incident light range from −11.867 degrees to −18.464 degrees. Relative to the second reflective axis, internal angles of incidence of the second incident light range from −11.670 degrees to −18.368 degrees. Thus it can be said that each of the first incident light and second incident light is offset from the first reflective axis by at least 11.670 degrees. In embodiments, incident light may be offset from its reflective axis by an internal angle of at least 1.0 degree; by at least 2.0 degrees; by at least 5.0 degrees; or by at least 9.0 degrees. A skew mirror or other reflective device configured to reflect incident light that is offset from the incident light's reflective axis can be advantageous in some applications. For example, in a head mounted display it may be advantageous to reflect an image toward a user's eye, but not to retroreflect the image back toward its source. Such reflection toward a user's eye typically requires that incident light be offset from its reflective axis by an internal angle of at least 5.0 degrees, and more typically by at least 9.0 degrees. Similarly, a device utilizing total internal reflection typically requires that incident light be offset from its reflective axis.

First embodiment external angles relative to surface normal for incident light and its reflection are also illustrated in FIGS. 11A and 11B. As seen in FIG. 11A, external angles relative to surface normal for first incident light 1124A, 1124B ranges from first incident light external angle 1113A of −7.000 degrees to first incident light external angle 1113B of +2.900 degrees. As seen in FIG. 11B, external angles relative to surface normal for second incident light 1130A, 1130B ranges from second incident light external angle 1115A of −7.000 to second incident light external angle 1115B of +2.900 degrees. First reflected light external angles 1114A, 1114B and second reflected light external angles 1116A, 1116B are also illustrated in FIGS. 11A and 11B, respectively. External angles are measured with the skew mirror residing in air, with refraction occurring at the skew mirror/air boundary. Angles of incidence and angles of reflection, and reflective axis angles are tabulated in Tables 1 and 2.

The physical properties of the first embodiment enable it to reflect light having other wavelengths, and to reflect light incident upon the grating medium at other angles, about substantially constant reflective axes. For example, the first embodiment grating structure's reflective properties enable it to reflect light having a wavelength of 520.4 nm about reflective axes having a mean reflective axis angle of +13.726 degrees, where the reflective axis angles vary by 0.10 degree or less for angles of incidence ranging from −6.862 degrees to +13.726 degrees and all angles in between (a range of 20.588 degrees). In another example of its reflective properties, the first embodiment is configured to reflect incident light about reflective axes (having a mean reflective axis angle of +13.726°), where the reflective axis angles vary by 0.20 degree or less for wavelengths at 503 nm and 537 nm (a range of 34 nm, WF=0.065, including a continuous spectrum of wavelengths between 503 nm and 537 nm), where the angle of incidence (internal, relative to surface normal) is −1.174 degrees.

For clarity, light in FIGS. 11A and 11B is illustrated as being reflected at a point residing proximate a center of the grating structure 1105. However, persons skilled in the art recognize that light is typically reflected throughout the grating structure rather than at a specific point.

In some embodiments, the first incident light and the second incident light have wavelengths other than 532 and 513, respectively. Similarly, embodiments include first and second reflective axes that may be coincident with surface normal, or may differ from surface normal.

A Second Embodiment Skew Mirror

Inventive aspects of a second embodiment skew mirror include the mirror being configured to reflect incident light having a wavelength of 532 nm and incident light having a wavelength of 513 nm about reflective axes that collectively have a mean reflective axis angle of +14.62 degrees relative to surface normal. In a further inventive aspect, the mean reflective axis angle (+14.618 degrees) for 532 nm light incident upon the skew mirror at internal angles of incidence ranging from −9.281 to −2.665 degrees differs by less than 0.001 degree from the mean reflective axis angle (+14.617 degrees) for 513 nm light incident upon the skew mirror at the same angles of incidence as the 532 nm incident light. The reflective axes are thus substantially constant for the 532 nm to 513 nm wavelength range, a condition that obtains for internal angles of incidence (relative to surface normal) from −9.281 degrees to −2.665 degrees.

A second embodiment skew mirror 1200 is illustrated in FIGS. 12A and 12B. The second embodiment skew mirror 1200 comprises a grating structure 1205 (shown by diagonal hatch lines in FIGS. 12A and 12B) residing in a grating medium 1210. For purposes of clarity, the diagonal hatch lines are omitted in a region within the grating medium 1210 proximate figure elements indicating light, axes, and angles. However, persons skilled in the art will recognize that the grating structure 1205 typically occupies the region described above. The grating structure 1205 of the second embodiment includes multiple holograms that at least partially overlap with each other in the grating medium 1210. The multiple holograms of the second embodiment comprise forty nine (49) volume holograms, recorded with recording beams having a wavelength of 405 nm. The 49 volume holograms overlap each other in the grating medium 1210, and are recorded in a manner similar to the first embodiment skew mirror, except that recording beam internal angles of incidence are adjusted to account for media shrinkage. Recording the 49 holograms of the second embodiment skew mirror is described below in a second method of making a skew mirror.

The second embodiment grating medium 1210 is a proprietary photosensitive polymeric optical recording medium, designated AK233-200, available from Akonia Holographics, LLC (Longmont, Colo.). The AK233-200 recording medium of the second embodiment is approximately 200 μm thick, has an M/# of approximately 24, and a refractive index of approximately 1.50 for light having a wavelength of 405 nm. The AK233-200 medium typically shrinks about 0.50% as a result of recording volume holograms.

Variations of the second embodiment skew mirror 1200 may include an additional layer such as a glass cover or glass substrate (not shown in FIGS. 12A and 12B). The additional layer is typically refractive index matched to the grating medium, and a thin film of index matching fluid may reside between the grating medium 1210 and the additional layer.

As shown in FIG. 12A, the grating structure 1205 of the second embodiment has the physical property of being configured to reflect a first incident light 1224A, 1224B, about a first reflective axis 1238 (shown in broken line). The first incident light has a first wavelength of 532 nm and is incident upon the grating medium 1210 at a specific site 1217. The first reflective axis 1238 differs from surface normal 1222 of the grating medium by a first reflective axis angle 1235 of +14.618 degrees (internal) relative to surface normal, where the first incident light has a first internal angle of incidence 1225A, 1225B, relative to surface normal, residing between −9.281 degrees to −2.665 degrees, inclusive (a range of 6.616 degrees). The first internal angle of incidence includes one hundred one (101) different internal angles spaced at angle intervals of approximately 0.066 degrees, from −9.281 degrees to −2.665 degrees. In some variations of the second embodiment skew mirror, the first internal angles of incidence for the first incident light include ten (10) different internal angles spaced at angle intervals of about 0.66 degrees, from −9.281 degrees to −2.665 degrees.

As shown FIG. 12A, first incident light 1224A, having a first internal angle of incidence 1225A of −9.281 degrees relative to surface normal, is reflected by the grating structure 1205 as first reflected light 1227A, having a first internal angle of reflectance 1226A of +38.610 degrees relative to surface normal. First incident light 1224B, having a first internal angle of incidence 1225B relative to surface normal of −2.665 degrees, is reflected as first reflected light 1227B having a first internal angle of reflectance 1226B of +31.836 degrees. First reflected light 1224A, 1224B has the first wavelength, i.e. in the second embodiment the first reflected light has a wavelength of 532 nm. First incident light angles, first reflected light angles, and first reflective axis angles, for the second embodiment skew mirror are shown in Table 3.

TABLE 3 ANGLES OF FIRST INCIDENT LIGHT, FIRST REFLECTED LIGHT, AND FIRST REFLECTIVE AXIS, FOR A SECOND EMBODIMENT SKEW MIRROR; WAVELENGTH = 532 nm; AK233-200 RECORDING MEDIUM; N = 101 Angle of Angle Of Angle Of Angle of Reflection First First Incidence Reflection Incidence of First Internal First Internal of First of First of First Reflected First Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 31.836 14.585 −2.665 −17.250 17.250 −4.000 52.300 24.150 31.941 14.604 −2.732 −17.336 17.336 −4.100 52.520 24.210 31.998 14.600 −2.799 −17.398 17.398 −4.200 52.640 24.220 32.103 14.619 −2.865 −17.484 17.484 −4.299 52.861 24.281 32.160 14.614 −2.932 −17.546 17.546 −4.400 52.980 24.290 32.217 14.610 −2.998 −17.607 17.607 −4.499 53.101 24.301 32.321 14.628 −3.065 −17.693 17.693 −4.600 53.320 24.360 32.378 14.623 −3.131 −17.754 17.754 −4.699 53.441 24.371 32.433 14.618 −3.198 −17.816 17.816 −4.800 53.560 24.380 32.490 14.613 −3.264 −17.877 17.877 −4.899 53.681 24.391 32.546 14.607 −3.331 −17.938 17.938 −5.000 53.800 24.400 32.602 14.603 −3.397 −18.000 18.000 −5.099 53.921 24.411 32.704 14.620 −3.464 −18.084 18.084 −5.200 54.140 24.470 32.760 14.615 −3.530 −18.145 18.145 −5.299 54.261 24.481 32.815 14.609 −3.597 −18.206 18.206 −5.400 54.380 24.490 32.871 14.604 −3.664 −18.267 18.267 −5.500 54.500 24.500 32.926 14.598 −3.730 −18.328 18.328 −5.600 54.620 24.510 33.027 14.616 −3.796 −18.412 18.412 −5.699 54.841 24.571 33.082 14.609 −3.863 −18.472 18.472 −5.800 54.960 24.580 33.137 14.604 −3.929 −18.533 18.533 −5.899 55.081 24.591 33.191 14.598 −3.996 −18.594 18.594 −6.000 55.200 24.600 33.291 14.615 −4.062 −18.677 18.677 −6.099 55.421 24.661 33.345 14.608 −4.129 −18.737 18.737 −6.200 55.540 24.670 33.400 14.603 −4.195 −18.797 18.797 −6.299 55.661 24.681 33.498 14.618 −4.262 −18.880 18.880 −6.400 55.880 24.740 33.552 14.612 −4.327 −18.940 18.940 −6.499 56.001 24.751 33.605 14.605 −4.395 −19.000 19.000 −6.600 56.120 24.760 33.659 14.600 −4.460 −19.060 19.060 −6.699 56.241 24.771 33.757 14.615 −4.527 −19.142 19.142 −6.800 56.460 24.830 33.810 14.608 −4.593 −19.201 19.201 −6.899 56.580 24.841 33.862 14.601 −4.660 −19.261 19.261 −7.000 56.699 24.850 33.916 14.595 −4.726 −19.321 19.321 −7.099 56.820 24.861 34.012 14.609 −4.793 −19.402 19.402 −7.200 57.039 24.920 34.064 14.603 −4.859 −19.462 19.462 −7.299 57.160 24.931 34.116 14.595 −4.926 −19.521 19.521 −7.400 57.279 24.940 34.169 14.588 −4.992 −19.580 19.580 −7.500 57.399 24.950 34.264 14.603 −5.058 −19.661 19.661 −7.600 57.619 25.010 34.316 14.596 −5.124 −19.720 19.720 −7.699 57.740 25.021 34.368 14.588 −5.191 −19.779 19.779 −7.800 57.860 25.030 34.462 14.602 −5.257 −19.860 19.860 −7.900 58.080 25.090 34.513 14.595 −5.324 −19.918 19.918 −8.000 58.199 25.100 34.606 14.608 −5.390 −19.998 19.998 −8.100 58.419 25.160 34.699 14.622 −5.456 −20.078 20.078 −8.200 58.639 25.220 34.750 14.614 −5.522 −20.136 20.136 −8.299 58.760 25.231 34.842 14.626 −5.589 −20.216 20.216 −8.401 58.978 25.289 34.893 14.619 −5.655 −20.274 20.274 −8.500 59.100 25.300 34.943 14.611 −5.721 −20.332 20.332 −8.600 59.220 25.310 35.035 14.624 −5.787 −20.411 20.411 −8.699 59.441 25.371 35.084 14.615 −5.854 −20.469 20.469 −8.800 59.560 25.380 35.134 14.607 −5.919 −20.527 20.527 −8.899 59.681 25.391 35.224 14.619 −5.986 −20.605 20.605 −9.000 59.900 25.450 35.273 14.611 −6.052 −20.662 20.662 −9.099 60.021 25.461 35.321 14.601 −6.119 −20.720 20.720 −9.200 60.140 25.470 35.411 14.613 −6.184 −20.798 20.798 −9.299 60.361 25.531 35.459 14.604 −6.251 −20.855 20.855 −9.400 60.479 25.540 35.548 14.616 −6.316 −20.932 20.932 −9.499 60.700 25.601 35.595 14.606 −6.383 −20.989 20.989 −9.600 60.819 25.610 35.683 14.617 −6.449 −21.066 21.066 −9.699 61.040 25.671 35.731 14.608 −6.516 −21.123 21.123 −9.800 61.159 25.680 35.817 14.618 −6.582 −21.200 21.200 −9.900 61.379 25.740 35.865 14.608 −6.648 −21.256 21.256 −10.000 61.499 25.750 35.951 14.618 −6.714 −21.332 21.332 −10.100 61.719 25.810 35.997 14.609 −6.780 −21.389 21.389 −10.200 61.839 25.820 36.083 14.619 −6.845 −21.464 21.464 −10.299 62.060 25.881 36.168 14.628 −6.912 −21.540 21.540 −10.400 62.279 25.940 36.214 14.618 −6.977 −21.596 21.596 −10.499 62.400 25.951 36.298 14.627 −7.044 −21.671 21.671 −10.600 62.619 26.010 36.343 14.617 −7.110 −21.726 21.726 −10.699 62.739 26.020 36.426 14.625 −7.176 −21.801 21.801 −10.800 62.958 26.079 36.471 14.615 −7.242 −21.856 21.856 −10.899 63.079 26.090 36.553 14.623 −7.308 −21.931 21.931 −11.000 63.298 26.149 36.635 14.631 −7.373 −22.004 22.004 −11.099 63.519 26.210 36.679 14.620 −7.440 −22.060 22.060 −11.200 63.638 26.219 36.761 14.628 −7.505 −22.133 22.133 −11.299 63.859 26.280 36.804 14.616 −7.572 −22.188 22.188 −11.400 63.978 26.289 36.885 14.624 −7.637 −22.261 22.261 −11.499 64.199 26.350 36.964 14.630 −7.704 −22.334 22.334 −11.600 64.418 26.409 37.007 14.619 −7.769 −22.388 22.388 −11.699 64.539 26.420 37.086 14.625 −7.836 −22.461 22.461 −11.800 64.758 26.479 37.164 14.631 −7.901 −22.533 22.533 −11.900 64.978 26.539 37.241 14.637 −7.967 −22.604 22.604 −12.000 65.198 26.599 37.284 14.625 −8.033 −22.658 22.658 −12.100 65.318 26.609 37.360 14.630 −8.099 −22.729 22.729 −12.200 65.537 26.669 37.436 14.636 −8.165 −22.800 22.800 −12.300 65.757 26.729 37.512 14.640 −8.231 −22.871 22.871 −12.400 65.977 26.789 37.553 14.629 −8.296 −22.924 22.924 −12.499 66.098 26.800 37.627 14.633 −8.362 −22.995 22.995 −12.600 66.317 26.859 37.702 14.637 −8.427 −23.064 23.064 −12.699 66.538 26.920 37.774 14.640 −8.494 −23.134 23.134 −12.800 66.756 26.978 37.848 14.645 −8.559 −23.203 23.203 −12.899 66.978 27.040 37.920 14.648 −8.625 −23.273 23.273 −13.000 67.197 27.099 37.960 14.635 −8.690 −23.325 23.325 −13.099 67.318 27.110 38.031 14.637 −8.756 −23.394 23.394 −13.200 67.537 27.169 38.102 14.640 −8.822 −23.462 23.462 −13.300 67.757 27.229 38.172 14.642 −8.888 −23.530 23.530 −13.400 67.977 27.289 38.242 14.644 −8.953 −23.597 23.597 −13.499 68.197 27.349 38.310 14.645 −9.019 −23.664 23.664 −13.600 68.415 27.408 38.379 14.647 −9.084 −23.731 23.731 −13.699 68.636 27.469 38.446 14.648 −9.150 −23.798 23.798 −13.800 68.855 27.528 38.514 14.649 −9.215 −23.864 23.864 −13.899 69.076 27.589 38.610 14.664 −9.281 −23.946 23.946 −14.000 69.395 27.698 Mean = 14.618 Mean = 25.594 Std. 0.016 Dev. =

As shown in FIG. 12B, the grating structure 1205 of the second embodiment is further configured to reflect second incident light 1230A, 1230B about a second reflective axis 1239. The second incident light has a second wavelength of 513 nm, and the second wavelength therefore differs from the first wavelength by 19 nm, or a wave fraction (WF) of 0.036. The second incident light is incident upon the grating medium 1210 at the specific site 1217. The specific site 1217 of the second embodiment includes an area of the grating medium surface 1212 upon which both the first and second incident light shine. The second reflective axis 1239 differs from surface normal 1222 of the grating medium by a second reflective axis angle 1236 of +14.617 degrees (internal) relative to surface normal, where the second incident light has a second internal angle of incidence 1228A, 1228B relative to surface normal, spanning a range of −9.281 degrees to −2.665 degrees. The second internal angle of incidence of the second incident light includes one hundred one (101) different internal angles spaced at angle intervals of approximately 0.066 degrees, from −9.281 degrees to −2.665 degrees. In some variations of the second embodiment skew mirror, the second internal angles of incidence for the second incident light include ten (10) different internal angles spaced at angle intervals of about 0.66 degrees, from −9.281 degrees to −2.665 degrees.

As shown in FIG. 12B, second incident light 1230A, having a second internal angle incidence 1228A of −9.281 degrees relative to surface normal, is reflected by the grating structure 1205 as second reflected light 1233A, having a second internal angle of reflectance 1229A of +38.598 degrees relative to surface normal. Second incident light 1230B, having a second internal angle of incidence 1228B relative to surface normal of −2.655 degrees, is reflected as second reflected light 1233B having a second internal angle of reflectance 1229B of +31.836 degrees. Second reflected light 1233A, 1233B has the second wavelength, i.e. in the second embodiment the second reflected light has a wavelength of 513 nm. Second incident light angles, second reflected light angles, and second reflective axis angles for the second embodiment skew mirror 1200 are shown in Table 4.

TABLE 4 ANGLES OF SECOND INCIDENT LIGHT, SECOND REFLECTED LIGHT, AND SECOND REFLECTIVE AXIS, FOR A SECOND EMBODIMENT SKEW MIRROR; WAVELENGTH = 513 nm; AK233-200 RECORDING MEDIUM; N = 101 Angle of Angle Of Angle Of Angle of Reflection Second Second Incidence Reflection Incidence of Second Internal Second Internal of Second of Second of Second Reflected Second Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 31.836 14.585 −2.665 −17.250 17.250 −4.000 52.300 24.150 31.941 14.604 −2.732 −17.336 17.336 −4.100 52.520 24.210 32.022 14.612 −2.799 −17.410 17.410 −4.200 52.690 24.245 32.080 14.608 −2.865 −17.472 17.472 −4.299 52.811 24.256 32.160 14.614 −2.932 −17.546 17.546 −4.400 52.980 24.290 32.240 14.621 −2.998 −17.619 17.619 −4.499 53.150 24.326 32.297 14.616 −3.065 −17.681 17.681 −4.600 53.270 24.335 32.378 14.623 −3.131 −17.754 17.754 −4.699 53.441 24.371 32.434 14.618 −3.198 −17.816 17.816 −4.800 53.561 24.381 32.514 14.625 −3.264 −17.889 17.889 −4.899 53.732 24.417 32.570 14.619 −3.331 −17.950 17.950 −5.000 53.851 24.426 32.626 14.615 −3.397 −18.011 18.011 −5.099 53.972 24.437 32.705 14.620 −3.464 −18.084 18.084 −5.200 54.141 24.471 32.737 14.604 −3.530 −18.134 18.134 −5.299 54.212 24.457 32.816 14.610 −3.597 −18.207 18.207 −5.400 54.382 24.491 32.872 14.605 −3.663 −18.267 18.267 −5.500 54.503 24.502 32.950 14.610 −3.730 −18.340 18.340 −5.600 54.672 24.536 33.006 14.605 −3.796 −18.401 18.401 −5.699 54.794 24.548 33.060 14.598 −3.863 −18.461 18.461 −5.800 54.912 24.556 33.137 14.604 −3.929 −18.533 18.533 −5.899 55.082 24.592 33.215 14.609 −3.996 −18.605 18.605 −6.000 55.252 24.626 33.292 14.615 −4.062 −18.677 18.677 −6.099 55.423 24.662 33.346 14.608 −4.129 −18.737 18.737 −6.200 55.541 24.671 33.423 14.614 −4.195 −18.809 18.809 −6.299 55.713 24.707 33.477 14.608 −4.262 −18.869 18.869 −6.400 55.833 24.717 33.554 14.613 −4.327 −18.941 18.941 −6.499 56.004 24.753 33.607 14.606 −4.395 −19.001 19.001 −6.600 56.123 24.762 33.683 14.611 −4.460 −19.072 19.072 −6.699 56.294 24.798 33.758 14.615 −4.527 −19.143 19.143 −6.800 56.463 24.832 33.812 14.609 −4.593 −19.202 19.202 −6.899 56.584 24.843 33.886 14.613 −4.660 −19.273 19.273 −7.000 56.752 24.876 33.939 14.607 −4.726 −19.333 19.333 −7.099 56.874 24.888 33.992 14.599 −4.793 −19.392 19.392 −7.200 56.994 24.897 34.067 14.604 −4.859 −19.463 19.463 −7.299 57.165 24.933 34.141 14.608 −4.926 −19.533 19.533 −7.400 57.335 24.968 34.192 14.600 −4.992 −19.592 19.592 −7.500 57.454 24.977 34.266 14.604 −5.058 −19.662 19.662 −7.600 57.624 25.012 34.318 14.597 −5.124 −19.721 19.721 −7.699 57.745 25.023 34.391 14.600 −5.191 −19.791 19.791 −7.800 57.915 25.058 34.443 14.593 −5.257 −19.850 19.850 −7.900 58.036 25.068 34.258 14.467 −5.324 −19.791 19.791 −8.000 57.606 24.803 34.418 14.514 −5.390 −19.904 19.904 −8.100 57.977 24.939 34.576 14.560 −5.456 −20.016 20.016 −8.200 58.348 25.074 34.733 14.606 −5.522 −20.127 20.127 −8.299 58.719 25.210 34.846 14.629 −5.589 −20.217 20.217 −8.401 58.988 25.294 34.897 14.621 −5.654 −20.276 20.276 −8.500 59.109 25.305 34.967 14.623 −5.721 −20.344 20.344 −8.600 59.279 25.340 35.018 14.615 −5.787 −20.402 20.402 −8.699 59.400 25.351 35.108 14.627 −5.854 −20.481 20.481 −8.800 59.618 25.409 35.137 14.609 −5.919 −20.528 20.528 −8.899 59.690 25.396 35.207 14.610 −5.986 −20.596 20.596 −9.000 59.859 25.430 35.277 14.612 −6.052 −20.664 20.664 −9.099 60.030 25.466 35.345 14.613 −6.119 −20.732 20.732 −9.200 60.198 25.499 35.414 14.615 −6.184 −20.799 20.799 −9.299 60.368 25.535 35.482 14.615 −6.251 −20.866 20.866 −9.400 60.536 25.568 35.551 14.617 −6.316 −20.934 20.934 −9.499 60.708 25.605 35.618 14.617 −6.383 −21.001 21.001 −9.600 60.876 25.638 35.666 14.608 −6.449 −21.058 21.058 −9.699 60.996 25.649 35.753 14.619 −6.516 −21.134 21.134 −9.800 61.216 25.708 35.820 14.619 −6.582 −21.201 21.201 −9.900 61.385 25.743 35.887 14.619 −6.648 −21.267 21.267 −10.000 61.555 25.778 35.954 14.620 −6.713 −21.334 21.334 −10.100 61.727 25.814 36.020 14.620 −6.780 −21.400 21.400 −10.200 61.897 25.849 36.067 14.611 −6.845 −21.456 21.456 −10.299 62.017 25.859 36.170 14.629 −6.912 −21.541 21.541 −10.400 62.286 25.943 36.217 14.620 −6.977 −21.597 21.597 −10.499 62.407 25.954 36.282 14.619 −7.044 −21.663 21.663 −10.600 62.577 25.989 36.365 14.628 −7.110 −21.737 21.737 −10.699 62.798 26.050 36.429 14.627 −7.176 −21.803 21.803 −10.800 62.967 26.084 36.475 14.617 −7.242 −21.858 21.858 −10.899 63.089 26.095 36.557 14.625 −7.308 −21.933 21.933 −11.000 63.309 26.155 36.621 14.624 −7.373 −21.997 21.997 −11.099 63.480 26.191 36.665 14.612 −7.440 −22.053 22.053 −11.200 63.599 26.200 36.746 14.620 −7.505 −22.126 22.126 −11.299 63.819 26.260 36.826 14.627 −7.572 −22.199 22.199 −11.400 64.037 26.319 36.888 14.626 −7.637 −22.263 22.263 −11.499 64.209 26.355 36.950 14.623 −7.704 −22.327 22.327 −11.600 64.379 26.390 37.029 14.630 −7.769 −22.399 22.399 −11.699 64.600 26.451 37.107 14.636 −7.836 −22.472 22.472 −11.800 64.819 26.510 37.185 14.642 −7.901 −22.543 22.543 −11.900 65.039 26.570 37.228 14.630 −7.967 −22.598 22.598 −12.000 65.159 26.580 37.305 14.636 −8.033 −22.669 22.669 −12.100 65.380 26.640 37.364 14.633 −8.099 −22.731 22.731 −12.200 65.549 26.675 37.440 14.638 −8.165 −22.803 22.803 −12.300 65.770 26.735 37.499 14.634 −8.231 −22.865 22.865 −12.400 65.940 26.770 37.557 14.631 −8.296 −22.926 22.926 −12.499 66.111 26.806 37.632 14.635 −8.362 −22.997 22.997 −12.600 66.330 26.865 37.706 14.639 −8.427 −23.067 23.067 −12.699 66.551 26.926 37.779 14.643 −8.494 −23.136 23.136 −12.800 66.770 26.985 37.852 14.647 −8.559 −23.206 23.206 −12.899 66.991 27.046 37.908 14.641 −8.625 −23.266 23.266 −13.000 67.159 27.080 37.980 14.645 −8.690 −23.335 23.335 −13.099 67.380 27.141 38.051 14.647 −8.756 −23.404 23.404 −13.200 67.599 27.200 38.121 14.650 −8.822 −23.472 23.472 −13.300 67.819 27.260 38.176 14.644 −8.888 −23.532 23.532 −13.400 67.989 27.295 38.245 14.646 −8.953 −23.599 23.599 −13.499 68.208 27.355 38.314 14.647 −9.019 −23.666 23.666 −13.600 68.427 27.414 38.398 14.657 −9.084 −23.741 23.741 −13.699 68.697 27.499 38.465 14.657 −9.150 −23.808 23.808 −13.800 68.916 27.558 38.517 14.651 −9.215 −23.866 23.866 −13.899 69.087 27.594 38.598 14.658 −9.281 −23.940 23.940 −14.000 69.355 27.678 Mean = 14.617 Mean = 25.593 Std. 0.025 Dev. =

For clarity, light in FIGS. 12A and 12B is illustrated as being reflected at a point residing proximate a center of the grating structure 1205. However, persons skilled in the art recognize that light is typically reflected throughout the grating structure rather than at a specific point.

In the second embodiment, the second reflective axis angle differs from the first reflective axis angle by approximately 0.0005 degree across WF=0.036. This very low level of change can approach the level of precision of instrumentation used to measure reflection angles. Accordingly, for the purposes of the present invention, the second reflective axis can be said to not differ from the first reflective axis. For some applications, the difference between reflective axis angles should be 0.025 degree or less. For some other applications, the difference between reflective axis angles should be 0.010 degree or less across WF≥0.036. The second embodiment skew mirror meets these requirements. A Student's t-test (two-tailed) indicates no difference between the first reflective axis angle and the second reflective axis angle (N=101 per group; P=0.873). Moreover, a difference of 0.001 degree or less challenges the precision of instrumentation used to measure skew mirror reflection angles. Accordingly, for purposes of the present invention, where a second reflective axis differs from a first reflective axis by 0.001 degree or less, the second reflective axis can be said to not differ from the first reflective axis.

For the second embodiment skew mirror, angles of incidence of the first incident light vary from −17.250 degrees to −23.946 degrees relative to the first reflective axis. Angles of incidence of the second incident light relative to the second reflective axis vary from −17.250 degrees to −23.940 degrees. Thus it can be said that each of the first incident light and second incident light is offset from the first reflective axis by at least 17.20 degrees. For the second embodiment skew mirror, angles if incidence and angles of reflection relative to reflective axis, for incident light and its reflection, respectively, are tabulated in Tables 3 and 4.

Second embodiment external angles relative to surface normal for incident light and its reflection are also illustrated in FIGS. 12A and 12B. As seen in FIG. 12A, external angles relative to surface normal for first incident light 1224A, 1224B ranges from first incident light external angle 1213A of −14.000 degrees to first incident light external angle 1213B of −4.000 degrees. As seen in FIG. 12A, external angles relative to surface normal for second incident light 1230A, 1230B ranges from second incident light external angle 1215A of −14.000 to second incident light external angle 1215B of −4.000 degrees. First reflected light external angles 1214A, 1214B and second reflected light external angles 1216A, 1216B are also illustrated in FIGS. 12A and 12B, respectively.

Persons skilled in the art will recognize that incident light and its reflection can typically be reversed, such that what was previously an angle of reflection becomes and angle of incidence, and vice versa. However, for purposes of the present disclosure, recitation or description of a range of incidence angles refers to incident light oriented to one side or the other of a reflective axis, but not both, or in the case of retroreflected incident light, an incidence angle of zero (0) relative to the reflective axis. Accordingly, a range of incidence angles does not include angles that are both positive and negative with respect to the reflective axes. As illustrated and described here, incidence angles relative to their respective reflective axes are negative (i.e. in a clockwise direction). However, this convention is used for convenience and simplicity and is not meant to teach, suggest, or imply that a skew mirror can only reflect incident light residing to one side of a reflective axis.

A Third Embodiment Skew Mirror

A third embodiment skew mirror comprises a grating structure residing in a grating medium, wherein the grating structure comprises twenty one (21) volume holograms that overlap each other in the grating medium.

The third embodiment grating medium is a commercial photosensitive polymeric optical recording medium, designated BAYFOL® HX TP photopolymer film, available from Covestro AG (formerly Bayer MaterialScience AG) (Leverkusen, Germany) The BAYFOL® HX TP recording medium of the third embodiment is approximately 70 μm thick, and typically shrinks about 1.0% as a result of recording volume holograms. Accordingly, shrinkage compensation is typically employed when recording volume holograms in the third embodiment grating medium. Shrinkage compensation is described below in the method of making the third embodiment skew mirror.

Variations of the third embodiment skew mirror may include an additional layer such as a glass cover or glass substrate. The additional layer is typically refractive index matched to the grating medium, and a thin film of index matching fluid may reside between the third embodiment grating medium and the additional layer.

The grating structure of the third embodiment has the physical property of being configured to reflect a first incident light about a first reflective axis. The first incident light has a first wavelength of 532 nm and is incident upon the grating medium at a specific site. The first reflective axis differs from surface normal of the grating medium by a first reflective axis angle of +9.419 degrees (internal) relative to surface normal, where the first incident light has an internal angle, relative to surface normal, residing between −6.251 degrees and +0.334 degrees, inclusive (a range of 6.585 degrees). The internal angle of the first incident light includes multiple angles spanning a range of approximately 6.59 degrees, the multiple angles including one hundred (100) different internal angles spaced at angle intervals of approximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.

Third embodiment first incident light having an internal angle of −6.251 degrees relative to surface normal, is reflected by the grating structure as first reflected light having an internal angle of +25.027 degrees relative to surface normal. First incident light having an internal angle relative to surface normal of +0.334 degrees is reflected as first reflected light having an internal angle of +18.487 degrees. First reflected light has the first wavelength, i.e. in the third embodiment the first reflected light has a wavelength of 532 nm.

The grating structure of the third embodiment is further configured to reflect second incident light about a second reflective axis. The second incident light has a second wavelength of 513 nm, and the second wavelength therefor differs from the first wavelength by 19 nm, or a wave fraction (WF) of 0.036. The second incident light is incident upon the grating medium at the specific site. The second reflective axis differs from surface normal of the grating medium by a second reflective axis angle of +9.400 degrees (internal) relative to surface normal, where the second incident light has in internal angle, relative to surface normal, spanning a range from −6.251 degrees to +0.334 degrees. The internal angle of the second incident light includes one hundred (100) different internal angles spaced at angle intervals of approximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.

Third embodiment second incident light, having an internal angle of −6.251 degrees relative to surface normal, is reflected by the grating structure as second reflected light, having an internal angle of +24.967 degrees relative to surface normal. Second incident light having an internal angle relative to surface normal of +0.334 degrees is reflected as second reflected light having an internal angle of +18.425 degrees. Second reflected light has the second wavelength, i.e. in the third embodiment the second reflected light has a wavelength of 513 nm. The second reflective axis of the third embodiment is substantially coincident with the first reflective axis.

Tables 5 summarizes reflective properties of first, second, and third embodiment skew mirrors.

TABLE 5 DIFFERENCE BETWEEN REFLECTIVE AXIS ANGLES AT λ = 532 nm AND λ = 513 nm FIRST EMBODIMENT SECOND EMBODIMENT THIRD EMBODIMENT SKEW MIRROR SKEW MIRROR SKEW MIRROR (AK174-200 (AK233-200 (BAYFOL ® HX recording medium) recording medium) recording medium) N = 100 N = 101 N = 100 measurements measurements measurements Mean reflective axis 13.693° 14.617° 9.400° INTERNAL angle at λ = 532 nm * Mean reflective axis 13.759° 14.618° 9.419° INTERNAL angle at λ = 513 nm * Difference between 0.066° 0.0005° 0.018° reflective axis INTERNAL angle at λ = 532 nm and at λ = 513 nm** Incident Light INTERNAL −4.660° to +1.933° −9.281° to −2.665° −6.251° to +0.334° Angles *** (range = 6.593°) (range = 6.616°) (range = 6.585°) Mean reflective axis 22.234° 25.594° 14.720° EXTERNAL angle at λ = 532 nm * Mean reflective axis 22.110° 25.593° 14.690° EXTERNAL angle at λ = 513 nm * Difference between 0.124° 0.0005° 0.030° reflective axis EXTERNAL angle at λ = 532 nm and at λ = 513 nm** Incident Light EXTERNAL −7.000° to 2.900°  −14.000° to −4.000°  −9.400° to +0.501° Angles *** * mean angles are relative to surface normal, and are the means of N measurements at N incident light angles of incidence; both incident and reflected light have the specified wavelength (λ). **differences between mean reflective axis angles at λ = 532 nm and at λ = 513 nm are absolute values and thus excludes negative numbers. *** incident light angles of incidence, relative to surface normal.

The range of angles of incidence across which a reflective axis remains constant can by expressed in terms of MB As shown in Table 6 below, the reflective axis angle for the first embodiment skew mirror varies by less than 0.015 degree for incident light having a range of angles of incidence of ≥20×Δθ_(B), at multiple wavelengths that differ from each other by WF≥0.036. For the second embodiment skew mirror the reflective axis angle varies by less than 0.020 degree for incident light having a range of angles of incidence ≥20×Δθ_(B), at multiple wavelengths that differ from each other by WF≥0.036.

TABLE 6 CHANGE IN REFLECTIVE AXIS ANGLES ACROSS AN INCIDENCE ANGLE RANGE OF APPROXIMATELY 20 × Δθ_(B) Skew Mirror Difference In reflective Incident Light Embodiment λ * Axis Angles ** Angle Range *** Δθ_(B) ^(†) FIRST 532 nm 0.012° −3.198° to +0.400° 0.177° EMBODIMENT θ (3.598° = 20.3 × Δ_(B)) SKEW MIRROR (AK174-200 513 nm 0.012° −3.198° to +0.400° 0.171° recording medium) (3.598° = 21.0 × Δθ_(B)) SECOND 532 nm 0.019° −7.242° to −4.726° 0.126° EMBODIMENT (2.516° = 20.0 × Δθ_(B)) SKEW MIRROR (AK233-200 513 nm 0.016° −7.242° to −4.726° 0.122° recording medium) (2.516° = 20.6 × Δθ_(B)) * wavelength of both incident and reflected light. ** difference in reflective axis angles (internal, relative to surface normal) for incident light having a change in angle of incidence of approximately 20 × Δθ_(B). *** range of incident light angles of incidence (internal, relative to surface normal) approximately equal to 20 × Δθ_(B), for which the Difference In Reflective Axis Angles is reported in this table. ^(†)Δθ_(B) is calculated for an incident light angle of incidence at the midpoint of the Incident Light Angle Range reported in this table. Methods of Making a Skew Mirror

An exemplary system 1350 for making a skew mirror is illustrated in FIG. 13. The exemplary system 1350 includes a grating medium 1310 disposed between a first mirror 1352A and a second mirror 1352B. The first and second mirrors are arranged to direct a first recording beam 1354 and a second recording beam 1355 such that the recording beams intersect and interfere with each other to form an interference pattern that is recorded as a hologram 1305 in the grating medium 1310. The hologram 1305 is an example of a grating structure.

The recording beams may be referred to as a reference beam and a signal beam according to a convention sometimes used by persons skilled in the holographic arts. However, each of the first and second recording beams are typically monochromatic collimated plane wave beams that are identical to each other (except for angles at which they are incident upon the grating medium). Moreover, the so-called signal beam typically includes no data encoded therein that is not also present in the so-called reference beam. Thus designation of one recording beam as a signal beam and the other recording beam as a reference beam can be arbitrary, with the designation of “signal” and “reference” serving to distinguish between the two recording beams, rather than to indicate that the one recording beam includes encoded data not present in the other recording beam.

In some embodiments the recording beams may have widths that differ from each other, or they may be the same. The recording beams may each have the same intensity as each other, or intensity can differ among the beams. The grating medium 1310 is typically secured in place between a first prism 1359A and second prism 1359B using a fluid index matched to both the prisms and the grating medium. A skew axis 1361 resides at a skew angle 1364 relative to surface normal 1322. The first and second recording beams 1354, 1355 reside at a first recording beam internal angle 1356 and a second recording beam internal angle 1357, respectively, relative surface normal 1322. Beam difference angle (a) 1358 is an angle of the first and second recording beams 1354, 1355, relative to each other. In embodiments, a resides in a range from 0 to 180 degrees. The skew angle 1364 for each hologram can be calculated according to equation (9), θ_(skew)=(θ_(R1)+θ_(R2)−180°)/2  (9) where: θ_(skew) is the skew angle, i.e., the internal angle of the skew axis relative to surface normal;

θ_(R1) is the first recording beam internal angle relative to surface normal; and

θ_(R2) is the second recording beam internal angle relative to surface normal.

As can be seen in FIG. 13, the first and second recording beams 1354, 1355 are symmetrical about the skew axis 1361 such that the first recording beam internal angle relative to the skew axis 1366 plus the second recording beam internal angle relative to the skew axis 1367 is equal to 180 degrees. The internal angles of the first and second recording beams relative to the skew axis 1366, 1367 are readily calculated from the first and second recording beam internal angles 1356, 1357, respectively, and the skew angle 1364.

Each of the first and second recording beams are typically collimated plane wave beams originating from a laser light source. The plane wave beams may be illustrated using multiple light ray depictions for each recording beam. For clarity however, in FIG. 13 the first and second recording beams are illustrated using a single light ray depiction for each recording beam.

Refraction at air/prism boundaries, for example where the first recording beam 1354 intersects an air/prism boundary of the first prism 1359A and where the second recording beam 1355 intersects an air/prism boundary of the second prism 1359B, is shown figuratively rather than strictly quantitatively in FIG. 13. Because the prisms are typically index matched to the grating medium 1310, refraction at the prism/grating medium boundary can usually be ignored. In embodiments, the grating medium and prisms each have an index of refraction of approximately 1.50.

A skew angle for a hologram (including a mean skew angle for a collection of holograms) can be substantially identical to a reflective axis angle, meaning the skew angle or mean skew angle is within 1.0 degree of the reflective axis angle. Persons skilled in the art will recognize that the skew angle and reflective axis angle can be theoretically identical. However, due to limits in system precision and accuracy, shrinkage of recording medium that occurs during recording holograms, and other sources of error, the skew angle or mean skew angle as measured or estimated based on recording beam angles may not perfectly match the reflective axis angle as measured by incidence angles and reflection angles of light reflected by a skew mirror. Nevertheless, a skew angle determined based on recording beam angles can be within 1.0 degree of the reflective axis angle determined based on angles of incident light and its reflection, even where medium shrinkage and system imperfections contribute to errors in estimating skew angle and reflective axis angle. A skew axis/reflective axis is generally called a skew axis when referring to making a skew mirror (for example when describing recording a hologram in a skew mirror grating medium), and as a reflective axis when referring to light reflective properties of a skew mirror.

Angles at which the first and second recording beams 1354, 1355 are incident upon the grating medium are adjusted by rotating the first and second beam mirrors, 1352A, 1352B, respectively. Rotation of the beam mirrors, indicated by rotation arrows 1353, not only adjusts incidence angles, but would also change where in the grating medium 1310 the recording beams interfere with each other. Accordingly, when beam mirrors are rotated in order to adjust incidence angles, the grating medium 1310 and prisms 1359A, 1359B are moved translationally in order to record new holograms at approximately the same location in the grating medium as previously recorded holograms. Translation of the grating medium 1310 is indicated by translation arrow 1360.

In a variation of the exemplary system 1350, a variable wavelength laser is used to vary the wavelength of the first and second recording beams. Incidence angles of the first and second recording beams may be, but are not necessarily, held constant while the wavelength of the first and second recording beams is changed.

A First Method of Making a Skew Mirror

A first method of making a skew mirror is illustrated in FIG. 14. The skew mirror of the first method is the first embodiment skew mirror 1100, which is also illustrated in FIGS. 11A and 11B, and whose reflective properties are described above. The first method typically utilizes a system for making a skew mirror such as the exemplary system 1350 illustrated in FIG. 13 and described above. For clarity however, in FIG. 14 first and second prisms are omitted, and recording beams are illustrated without showing refraction at air/grating medium boundaries or air/prism boundaries. However, persons skilled in the art will recognize that refraction typically occurs at an air/prism boundary (or air/grating medium boundary, where index matched prisms are not used), and should be accounted for when designing a system or method to achieve the internal angles described.

A first recording beam 1154 and a second recording beam 1155 are directed at the first embodiment grating medium 1110, where the recording beams interfere with each other to create an interference pattern, which is recorded as a volume hologram in the grating medium 1110. The recording beams are typically created by splitting a 405 nm light beam from an external cavity, tunable diode laser into two separate beams. The light beam is split using a polarizing beam splitter, and a half wave plate is used to alter polarity of one of the two separate beams from p-polarized to s-polarized, such that both of the two separate beams are s-polarized. One of the s-polarized beams becomes the first recording beam 1154 and the other of the s-polarized beams becomes the second recording beam 1155. Each of the first and second recording beams is a collimated, plane wave beam having a wavelength of 405 nm.

The first embodiment skew mirror benefits from having reflective properties that allow it to reflect light at a substantially different wavelength, and in particular a considerably longer wavelength, than the recording beam wavelength. The AK174-200 grating medium, in which first embodiment holograms are recorded with 405 nm wavelength recording beams, absorbs 405 nm light at approximately 0.07 absorbance units for the 200 μm thick medium. Conversely, the AK174-200 grating medium has negligible absorbance for visible wavelengths of light greater than 425 nm (conservatively estimated at less than 0.002 absorbance units per 200 um; the negligible absorbance is typically indistinguishable from zero). Thus the AK174-200 grating medium absorbs recording beam light (at 405 nm) at least 35 times more strongly than green light (for example, in a range of 503 nm to 537 nm) the first embodiment skew mirror is configured to reflect.

The grating structure 1105 of the first embodiment skew mirror 1100 is created by recording 48 volume holograms in the grating medium 1110. Each of the 48 holograms is recorded at its own unique first recording beam internal angle 1156 and its own unique second recording beam internal angle 1157. The first recording beam internal angle 1156 is an internal angle of the first recording beam 1154 relative to surface normal 1122 of the grating medium 1110 and the second recording beam internal angle 1157 is an internal angle of the second recording beam 1155 relative to surface normal 1122. Beam difference angle (a) 1158 is an angle of the first and second recording beams 1154, 1155, relative to each other.

Each of the first and second recording beams for the first embodiment skew mirror has irradiance of approximately 3 mW/cm². Typically, the first of the 48 holograms is recorded with an energy dose of 35 mJ/cm², and the dose is increased by about 1.5% for each subsequent hologram. The total energy dose for recording all 48 holograms is typically about 2.5 J/cm². Irradiance and energy doses described here are merely exemplary. Other embodiments of skew mirrors and methods of making skew mirrors may use different levels of irradiance and energy dose.

A first hologram is recorded using a first recording beam internal angle 1156 of +53.218 degrees and a second recording beam internal angle 1157 of +154.234 degrees, resulting in a beam difference angle (α) 1158 of 101.016 degrees. The skew axis 1161 for each of the 48 holograms has a skew angle 1164 of +13.726 degrees relative to surface normal 1122, and the mean skew angle for the 48 holograms is therefore also +13.726 degrees. The skew angle for each hologram is calculated according to equation (9) above. For each subsequent hologram of the grating structure, the first and second recording beam internal angles 1156, 1157 are typically changed by amounts that are approximately equal in magnitude to each other but having opposite signs, which maintains symmetry of the first and second recording beams about the skew axis.

For example, for a second hologram, the first recording beam internal angle is changed by +0.091 degree and the second recording beam internal angle is adjusted by −0.091 degree, such that the first recording beam internal angle 1156 becomes +53.309 degrees and the second recording beam internal angle +154.143 degrees; α=100.834 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram vary slightly across the 48 volume holograms (i.e., the change in recording beam internal angles from one hologram to the next varies), from 0.091 degree for changes in recording beam internal angles from the first hologram to the second hologram, to 0.084 degree for changes in recording beam internal angles from the 47^(th) hologram to the 48^(th) hologram. However, for each change of first and second recording beam internal angles, the magnitude of change is the same and the sign is opposite for each of the first and second beam angles. The first and second recording beam internal angles 1156, 1157 for the last (48^(th)) hologram of the first embodiment grating structure 1105 are +57.332 and +150.120 degrees, respectively, and α=92.788 degrees. In some embodiments, the magnitude of change of the first recording beam internal angle may differ very slightly from the magnitude of change of the second recording beam internal angle, which may compensate for system imprecision, for Snell effects, for dispersion, or for shrinkage of the grating medium that results from recording the holograms.

Position of the grating medium 1110 is adjusted (as indicated by translation arrow 1160) between recording one hologram and the next hologram such that at least part of each of the 48 holograms is recorded in a common space in the grating medium shared with at least part of all others of the 48 holograms. Accordingly, each of the 48 holograms at least partially spatially overlaps all others of the 48 holograms in the grating medium.

The first recording beam internal angle 1156 ranges from +53.218 to +57.332 degrees (a range of 4.114 degrees) and the second recording beam internal angle 1157 ranges from +154.234 to +150.120 degrees (a range of 4.114 degrees). As can be seen in FIG. 14, for each hologram of the first method, the first and second recording beams 1154, 1155 are symmetrical about the skew axis 1161 such that the internal angle of the first recording beam relative to the skew axis 1166 (+38.492 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis 1167 (+141.508 degrees for the first hologram) is equal to 180 degrees (38.492°+141.508°=180°). The internal angles of the first and second recording beams relative to the skew axis 1166, 1167 are readily calculated from the first and second recording beam internal angles 1156, 1157, respectively, and the skew angle 1164. First and second recording beam internal angles (which are defined as internal angles relative to surface normal of the grating medium) and internal angles relative to the skew axis of the first and second recording beams are listed in Table 7. After recording the 48 volume holograms, the AK174-200 recording medium is light cured by a process familiar to persons skilled in the art. [need more here]

Beam difference angle α can be used to calculate K_(G) according to equations (10) and (11).

$\begin{matrix} {{K_{G} \equiv {{\overset{\rightharpoonup}{K}}_{G}}} = \sqrt{2{k^{2}\left( {1 - {\cos\mspace{11mu}\alpha}} \right)}}} & (10) \\ {{k \equiv {{\overset{\rightharpoonup}{k}}_{1}}} = {{{\overset{\rightharpoonup}{k}}_{2}} = \frac{2\pi\; n}{\lambda}}} & (11) \end{matrix}$

where: {right arrow over (K)}_(G) is a grating vector in k-space for a hologram, comprising both {right arrow over (K)}_(G+) and {right arrow over (K)}_(G−);

-   -   {right arrow over (k)}₁ and {right arrow over (k)}₂ are wave         vectors in k-space for first and second recording beams,         respectively, used for recording the hologram;     -   α is the beam difference angle as described above;     -   λ is the vacuum wavelength of light of the first and second         recording beams; and     -   n is the mean refractive index of the hologram (referred to as         the “bulk index,” for which the refractive index of the grating         medium in which the hologram is recorded is used as an         approximation).         A more detailed explanation of {right arrow over (K)}_(G) and         {right arrow over (k)} is found above in a section of this         specification titled k-Space Formalism for Holography.

For the AK174-200 grating medium used for the first embodiment skew mirror, n is estimated at 1.50 for light at 405 nm. The wavelength of the first and second recording beams used to record holograms for making the first embodiment skew mirror is 405 nm=405×10⁻⁹ m. Accordingly, k=2.327×10⁷ radians per meter (rad/m) for the first and second recording beams in AK174-200 grating medium. K_(G) for the holograms of the first embodiment skew mirror ranges from 3.592×10⁷ rad/m for the first hologram to 3.370×10⁷ rad/m for the 48^(th) hologram.

The absolute value of the difference in grating vectors |ΔK_(G)| between any two holograms can be a useful metric for describing hologram “spacing” (i.e. how close to each other in k-space are grating vectors for the any two holograms). |ΔK_(G)| for each first embodiment hologram and its adjacent hologram(s) is relatively constant, with a mean value for all 48 holograms of 4.715×10⁴ rad/m and a coefficient of variation of 0.11%. The adjacent hologram(s) for each hologram consist of a hologram or holograms having the next highest or the next lowest K_(G) compared to K_(G) for the each hologram. |ΔK_(G)| for each first embodiment hologram and its adjacent hologram(s) reside in a range between 4.70×10⁴ and less than 4.73×10⁴ rad/m. |ΔK_(G)| between the first and 48^(th) holograms is 2.22×10⁶ rad/m.

In skew mirror embodiments, |ΔK_(G)| between a hologram and an adjacent hologram (which can be referred to as adjacent |ΔK_(G)|) has a mean value for multiple holograms that typically, but not necessarily, resides in a range between 5.0×10³ and 1.0×10⁷ rad/m, more typically in a range between 1.0×10⁴ and 5×10⁶ rad/m, and more typically still in a range between 1.0×10⁴ and 1.0×10⁶. In some embodiments, mean adjacent |ΔK_(G)| for multiple holograms resides in a range between 8.0×10⁴ and 5.0×10⁶ rad/m, and may reside in a range between 1.0×10⁵ and 1.0×10⁶ rad/m.

In some embodiments, mean adjacent |ΔK_(G)| for multiple holograms strongly influences skew mirror performance Relatively small mean adjacent |ΔK_(G)| for a set of holograms can correspond to relatively high skew mirror image fidelity. However, where adjacent |ΔK_(G)| for a set of holograms is relatively small, the total number of holograms in the set is larger in order to span a given |ΔK_(G)| range for the hologram set. Moreover, given that recording capacity for grating mediums is typically limited by dynamic range (usually expressed as M/#), recording more holograms in a set usually means that each hologram in the set is weaker (i.e., is recorded more faintly in the medium). Accordingly, tension exists between having relatively small adjacent |ΔK_(G)| for a hologram set (which requires more holograms, other things being equal), and having a larger adjacent |ΔK_(G)| for the set, which enables recording fewer, but stronger holograms. Fewer, stronger holograms typically results in stronger reflectance by a skew mirror. Furthermore, the use of relatively broadband illumination sources (e.g., LEDs instead of lasers) may reduce image fidelity loss in a skew mirror with a larger mean adjacent |ΔK_(G)|. In some embodiments, a sweet spot exists where mean adjacent |ΔK_(G)| for multiple holograms resides in the range between 5.0×10³ rad/m and 1.0×10⁷ rad/m. Embodiments of skew mirrors where mean adjacent |ΔK_(G)| resides in the sweet spot typically exhibit a desirable balance of image fidelity and reflectance.

Values for α, K_(G), and |ΔK_(G)| for each of the 48 holograms of the first embodiment skew mirror can be found in Table 7.

TABLE 7 RECORDING BEAM ANGLES FOR A FIRST METHOD OF MAKING A SKEW MIRROR; SKEW ANGLE = 13.726° RELATIVE TO SURFACE NORMAL Internal Internal |ΔK_(G)| First Second Angle of Angle of Magnitude Between Recording Recording First Second of Angle Hologram Beam Beam Recording Recording Difference and Its Angle Angle Beam Beam From Preceding (internal, relative to Relative To Relative To Previous Adjacent surface normal, in Skew Axis Skew Axis Hologram α K_(G) Hologram # degrees) (degrees) (degrees) (degrees) (degrees) (×10⁷ rad/m) (×10⁴ rad/m) 1 53.218 154.234 39.492 140.508 101.016 3.592 2 53.309 154.143 39.583 140.417 0.091 100.833 3.587 4.709 3 53.400 154.052 39.674 140.326 0.091 100.652 3.582 4.704 4 53.491 153.961 39.765 140.235 0.091 100.470 3.578 4.713 5 53.581 153.871 39.855 140.145 0.091 100.289 3.573 4.708 6 53.672 153.780 39.946 140.054 0.090 100.109 3.568 4.709 7 53.762 153.690 40.036 139.964 0.090 99.928 3.563 4.704 8 53.852 153.600 40.126 139.874 0.090 99.748 3.559 4.713 9 53.942 153.510 40.216 139.784 0.090 99.568 3.554 4.707 10 54.031 153.421 40.305 139.695 0.090 99.389 3.549 4.709 11 54.121 153.331 40.395 139.605 0.090 99.210 3.545 4.710 12 54.210 153.242 40.484 139.516 0.089 99.031 3.540 4.712 13 54.300 153.152 40.574 139.426 0.089 98.853 3.535 4.713 14 54.389 153.063 40.663 139.337 0.089 98.674 3.530 4.707 15 54.478 152.974 40.752 139.248 0.089 98.496 3.526 4.715 16 54.567 152.885 40.841 139.159 0.089 98.319 3.521 4.710 17 54.655 152.797 40.929 139.071 0.089 98.142 3.516 4.711 18 54.744 152.708 41.018 138.982 0.089 97.965 3.512 4.719 19 54.832 152.620 41.106 138.894 0.088 97.788 3.507 4.712 20 54.920 152.532 41.194 138.806 0.088 97.612 3.502 4.713 21 55.008 152.444 41.282 138.718 0.088 97.436 3.497 4.714 22 55.096 152.356 41.370 138.630 0.088 97.260 3.493 4.715 23 55.184 152.268 41.458 138.542 0.088 97.085 3.488 4.708 24 55.271 152.181 41.545 138.455 0.088 96.909 3.483 4.716 25 55.359 152.093 41.633 138.367 0.087 96.734 3.479 4.717 26 55.446 152.006 41.720 138.280 0.087 96.560 3.474 4.717 27 55.533 151.919 41.807 138.193 0.087 96.386 3.469 4.710 28 55.620 151.832 41.894 138.106 0.087 96.211 3.464 4.718 29 55.707 151.745 41.981 138.019 0.087 96.038 3.460 4.718 30 55.794 151.658 42.068 137.932 0.087 95.864 3.455 4.718 31 55.881 151.571 42.155 137.845 0.087 95.691 3.450 4.718 32 55.967 151.485 42.241 137.759 0.086 95.518 3.446 4.711 33 56.053 151.399 42.327 137.673 0.086 95.346 3.441 4.719 34 56.139 151.313 42.413 137.587 0.086 95.173 3.436 4.719 35 56.225 151.227 42.499 137.501 0.086 95.001 3.431 4.718 36 56.311 151.141 42.585 137.415 0.086 94.829 3.427 4.718 37 56.397 151.055 42.671 137.329 0.086 94.658 3.422 4.718 38 56.483 150.969 42.757 137.243 0.086 94.486 3.417 4.718 39 56.568 150.884 42.842 137.158 0.086 94.315 3.413 4.725 40 56.654 150.798 42.928 137.072 0.085 94.145 3.408 4.717 41 56.739 150.713 43.013 136.987 0.085 93.974 3.403 4.717 42 56.824 150.628 43.098 136.902 0.085 93.804 3.398 4.724 43 56.909 150.543 43.183 136.817 0.085 93.634 3.394 4.723 44 56.994 150.458 43.268 136.732 0.085 93.464 3.389 4.715 45 57.079 150.373 43.353 136.647 0.085 93.295 3.384 4.722 46 57.163 150.289 43.437 136.563 0.085 93.126 3.380 4.721 47 57.248 150.204 43.522 136.478 0.085 92.956 3.375 4.728 48 57.332 150.120 43.606 136.394 0.084 92.788 3.370 4.719 MEAN 4.715 STDEV 0.0054 % CV 0.11

In a variation of the first method of making a skew mirror, a hologram is created by continuously and synchronously adjusting the first and second recording beam internal angles while maintaining the symmetry of the first and second recording beams about the skew axis as described above. Accordingly, a single hologram is recorded while the first recording beam is scanned from a first recording beam internal angle of +53.218 degrees to a first recording beam angle of +57.332 degrees. Simultaneously, the second recording beam is scanned from a second recording beam internal angle of +154.234 degrees to +150.120 degrees. Accordingly, a varies from 101.016 degrees to 92.788 degrees and K_(G) varies from 3.592×10⁷ rad/m to 3.370×10⁷ rad/m while the single hologram is being recorded. Position of the grating medium is adjusted while the single hologram is being recorded such that the single hologram is recorded in a relatively compact space in the grating medium, rather than being smeared across a relatively broad space as the recording beam angles change. The single hologram thus exhibits reflective properties very similar to the 48 discrete holograms recorded with 48 sets of unique first recording beam and second recording beam internal angles, and the total energy dose for recording the single hologram is typically about the same (2.5 J/cm²) as for the 48 holograms.

A Second Method of Making a Skew Mirror

A second method of making a skew mirror is described below. The skew mirror made by the second method is the second embodiment skew mirror 1200, which is also illustrated in FIGS. 12A and 12B, and whose reflective properties are described above.

The second method is identical to the first method except that first and second recording beam internal angles are different than with the first method, which gives the second embodiment skew mirror reflective properties that differ from those of the first embodiment. In the present example, the second method is performed using a grating medium (AK233-200) that differs from that of the first method (AK174-200). Like the first embodiment, the second embodiment skew mirror benefits from having reflective properties that allow it to reflect light at a substantially different wavelength, and in particular a considerably longer wavelength, than the recording beam wavelength.

The grating structure 1205 of the second embodiment skew mirror 1200 is created by recording 49 volume holograms in the grating medium 1210. Each of the 49 holograms of the second method is recorded at its own unique first recording beam internal angle and its own unique second recording beam internal angle. The first recording beam internal angle is an internal angle of the first recording beam relative to surface normal of the grating medium and the second recording beam internal angle is an internal angle of the second recording beam relative to surface normal. Each of the first and second recording beams for the first embodiment skew mirror has irradiance of approximately 3 mW/cm². Typically, the first of the 49 holograms is recorded with an energy dose of 35 mJ/cm², and the dose is increased by about 1.5% for each subsequent hologram. The total dose for recording all 49 holograms is typically about 2.5 J/cm².

According to the second method, a first hologram is recorded using a first recording beam internal angle of +55.913 degrees and a second recording beam internal angle of +153.323 degrees; a therefore is 97.410 degrees. The skew axis for each of the 49 holograms has a skew angle of +14.618 degrees relative to surface normal. The skew angle for each hologram is calculated according to equation (9) above. For each subsequent hologram of the grating structure, the first and second recording beam internal angles are typically changed by amounts that are approximately equal in magnitude to each other, but having opposite signs, which maintains symmetry of the first and second recording beams about the skew axis.

For example, for recording a second hologram according to the second method, the first recording beam internal angle is changed by +0.095 degree and the second recording beam internal angle is adjusted by −0.095 degree, such that the first recording beam internal angle becomes +56.008 degrees and the second recording beam internal angle +153.228 degrees; α=97.220 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 49 volume holograms (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of 0.095 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of 0.087 degree for changes in recording beam internal angles from the 48^(th) hologram to the 49^(th) hologram. However, the magnitude of change is the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (49^(th)) hologram of the second embodiment grating structure are +60.252 and +148.984 degrees, respectively, and α=88.732. In some embodiments, the magnitude of change of the first recording beam internal angle may differ very slightly from the magnitude of change of the second recording beam internal angle in order to compensate for factors such as system imprecision, Snell effects, dispersion, or shrinkage of the grating medium that results from recording the holograms.

The position of the grating medium is adjusted between recording one hologram and the next such that at least part of each of the 49 holograms is recorded in a common space shared with at least part of all others of the 49 holograms. Accordingly, each of the 49 holograms at least partially spatially overlaps all others of the 49 holograms in the grating medium.

Thus according to the second method first recording beam internal angles range from +55.913 to +60.252 degrees (a range of 4.339 degrees) and the second recording beam internal angles range from +153.323 to +148.984 degrees (a range of 4.339 degrees). As with the first method, for each hologram of the second method the first and second recording beams are symmetrical about the skew axis such that the internal angle of the first recording beam relative to the skew axis (+41.295 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+138.705 for the first hologram)=180 degrees (41.295°+138.705°=180°). The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles relative to surface normal, respectively, and the skew angle. For the second method of making a skew mirror, first and second recording beam internal angles (which are defined as internal angles relative to surface normal of the grating medium) and internal angles relative to the skew axis for the first and second recording beams are listed in Table 8. After recording the 49 volume holograms, the AK233-200 recording medium is light cured by a process familiar to persons skilled in the art. For example, in some embodiments light curing comprises exposure to near-ultraviolet uniform coherent light from a light emitting diode, until substantially all photoinitiator, photoactive monomer, of other photoactive chemistry has been consumed.

TABLE 8 RECORDING BEAM ANGLES FOR A SECOND METHOD OF MAKING A SKEW MIRROR; SKEW ANGLE = 14.618° RELATIVE TO SURFACE NORMAL Internal Internal First Second Angle of Angle of Magnitude Recording Recording First Second of Angle Beam Beam Recording Recording Difference Angle Angle Beam Beam From (internal, relative to Relative To Relative To Previous surface normal, in Skew Axis Skew Axis Hologram α K_(G) ΔK_(G) # degrees) (degrees) (degrees) (degrees) (degrees) (×10⁷ rad/m) (×10⁴ rad/m) 1 55.913 153.323 41.295 138.705 97.410 3.497 2 56.008 153.228 41.390 138.610 0.095 97.220 3.492 5.098 3 56.102 153.134 41.484 138.516 0.094 97.032 3.487 5.053 4 56.196 153.040 41.578 138.422 0.094 96.844 3.482 5.063 5 56.290 152.946 41.672 138.328 0.094 96.656 3.477 5.072 6 56.384 152.852 41.766 138.234 0.094 96.468 3.471 5.081 7 56.477 152.759 41.859 138.141 0.093 96.282 3.466 5.037 8 56.571 152.665 41.953 138.047 0.094 96.094 3.461 5.100 9 56.664 152.572 42.046 137.954 0.093 95.908 3.456 5.055 10 56.757 152.479 42.139 137.861 0.093 95.722 3.451 5.064 11 56.849 152.387 42.231 137.769 0.092 95.538 3.446 5.019 12 56.942 152.294 42.324 137.676 0.093 95.352 3.441 5.082 13 57.034 152.202 42.416 137.584 0.092 95.168 3.436 5.036 14 57.127 152.109 42.509 137.491 0.093 94.982 3.431 5.100 15 57.219 152.017 42.601 137.399 0.092 94.798 3.426 5.054 16 57.311 151.925 42.693 137.307 0.092 94.614 3.421 5.063 17 57.402 151.834 42.784 137.216 0.091 94.432 3.416 5.017 18 57.494 151.742 42.876 137.124 0.092 94.248 3.411 5.081 19 57.585 151.651 42.967 137.033 0.091 94.066 3.406 5.034 20 57.676 151.560 43.058 136.942 0.091 93.884 3.401 5.043 21 57.767 151.469 43.149 136.851 0.091 93.702 3.396 5.051 22 57.858 151.378 43.240 136.760 0.091 93.520 3.391 5.060 23 57.949 151.287 43.331 136.669 0.091 93.338 3.385 5.068 24 58.040 151.196 43.422 136.578 0.091 93.156 3.380 5.077 25 58.130 151.106 43.512 136.488 0.090 92.976 3.375 5.029 26 58.220 151.016 43.602 136.398 0.090 92.796 3.370 5.038 27 58.310 150.926 43.692 136.308 0.090 92.616 3.365 5.046 28 58.400 150.836 43.782 136.218 0.090 92.436 3.360 5.054 29 58.490 150.746 43.872 136.128 0.090 92.256 3.355 5.063 30 58.579 150.657 43.961 136.039 0.089 92.078 3.350 5.015 31 58.669 150.567 44.051 135.949 0.090 91.898 3.345 5.079 32 58.758 150.478 44.140 135.860 0.089 91.720 3.340 5.031 33 58.847 150.389 44.229 135.771 0.089 91.542 3.335 5.039 34 58.936 150.300 44.318 135.682 0.089 91.364 3.330 5.047 35 59.025 150.211 44.407 135.593 0.089 91.186 3.325 5.055 36 59.113 150.123 44.495 135.505 0.088 91.010 3.320 5.006 37 59.202 150.034 44.584 135.416 0.089 90.832 3.315 5.071 38 59.290 149.946 44.672 135.328 0.088 90.656 3.310 5.022 39 59.378 149.858 44.760 135.240 0.088 90.480 3.305 5.030 40 59.466 149.770 44.848 135.152 0.088 90.304 3.300 5.037 41 59.554 149.682 44.936 135.064 0.088 90.128 3.295 5.045 42 59.642 149.594 45.024 134.976 0.088 89.952 3.290 5.053 43 59.730 149.506 45.112 134.888 0.088 89.776 3.285 5.061 44 59.817 149.419 45.199 134.801 0.087 89.602 3.280 5.011 45 59.904 149.332 45.286 134.714 0.087 89.428 3.275 5.018 46 59.991 149.245 45.373 134.627 0.087 89.254 3.270 5.026 47 60.078 149.158 45.460 134.540 0.087 89.080 3.264 5.033 48 60.165 149.071 45.547 134.453 0.087 88.906 3.259 5.041 49 60.252 148.984 45.634 134.366 0.087 88.732 3.254 5.048 MEAN 5.050 STDEV 0.0235 % CV 0.47

For the AK233-200 grating medium used for the second embodiment skew mirror, n is estimated at 1.50 for light at 405 nm. The wavelength of the first and second recording beams used to record holograms for making the second embodiment skew mirror is 405 nm=405×10⁻⁹ m. Accordingly, k=2.327×10⁷ rad/m for the first and second recording beams in AK233-200 grating medium. K_(G) for the holograms of the second embodiment skew mirror ranges from 3.497×10⁷ rad/m for the first hologram to 3.254×10⁷ rad/m for the 49^(th) hologram.

Adjacent |ΔK_(G)| for the second embodiment holograms is relatively constant, with a mean value for all 49 holograms of 5.050×10⁴ rad/m and a coefficient of variation of 0.47%. Adjacent |ΔK_(G)| for each second embodiment hologram resides in a range between 5.01×10⁴ to 5.10×10⁴ rad/m. |ΔK_(G)| between the first and 49^(th) holograms is 2.42×10⁶ rad/m.

In a variation of the second method of making a skew mirror, a hologram is created by continuously and synchronously adjusting the first and second recording beam internal angles while maintaining the symmetry of the first and second recording beams about the skew axis as described above. Accordingly, a single hologram is recorded while the first recording beam is scanned from a first recording beam internal angle of +55.913 degrees to a first recording beam angle of +60.252 degrees. Simultaneously, the second recording beam is scanned from a second recording beam internal angle of +153.323 degrees to +148.984 degrees. The single hologram is thus equivalent to the 49 discrete holograms recorded with 49 sets of unique first recording beam and second recording beam internal angles. The total energy dose for recording the single hologram is typically 2.5 J/cm² for the single hologram.

A Third Method of Making a Skew Mirror

A third method of making a skew mirror is described below. Like the first method, the third method typically utilizes a system for making a skew mirror such as the exemplary system 1350 illustrated in FIG. 13 and described above.

According to the third method, a grating structure is created by recording three sets of holograms in a grating medium. The first hologram set includes 21 holograms, the second hologram set includes 19 holograms, and the third hologram set includes 16 holograms, for a total of 56 holograms. In some embodiments, each of the first, second, and third holograms sets includes at least 6 holograms, or at least 9 holograms. Each of the multiple holograms in the first hologram set typically at least partially spatially overlaps at least one other of the multiple holograms in the first hologram set, and at least one of the multiple holograms in the first hologram set may at least partially spatially overlap at least one of the holograms in the second hologram set. In some embodiments, each of the multiple holograms in the first hologram set at least partially spatially overlaps all others of the multiple holograms in the first hologram set.

Similarly, each of the multiple holograms in the second hologram set typically at least partially spatially overlaps at least one other of the multiple holograms in the second hologram set, and at least one of the multiple holograms in the second hologram set may at least partially spatially overlap at least one of the holograms in the first hologram set or the third hologram set. In some embodiments, each of the multiple holograms in the second hologram set at least partially spatially overlaps all others of the multiple holograms in the second hologram set.

Similarly, each of the multiple holograms in the third hologram set typically at least partially spatially overlaps at least one other of the multiple holograms in the third hologram set, and at least one of the multiple holograms in the third hologram set may at least partially spatially overlap at least one of the holograms in the second hologram set. In some embodiments, each of the multiple holograms in the third hologram set at least partially spatially overlaps all others of the multiple holograms in the third hologram set. In some embodiments, all holograms of the first, second, and third hologram sets at least partially spatially overlap with each other.

Each of the 56 total holograms is recorded using first and second recording beams, each of which is incident upon the grating medium at its own unique first recording beam internal angle and its own unique second recording beam internal angle. In some embodiments, not every first and second recording beam internal angle is unique. For example, in some embodiments multiple holograms having the same recording beam internal angles as each other may be written in in locations in the skew mirror that differ from each other. The first recording beam internal angle is an internal angle of the first recording beam relative to surface normal of the grating medium and the second recording beam internal angle is an internal angle of the second recording beam relative to surface normal. Each of the first and second recording beams for the first embodiment skew mirror is a monochromatic collimated light beam having irradiance of approximately 3 mW/cm². Typically, the first of the 56 holograms is recorded with an energy dose of 35 mJ/cm², and the dose is increased by about 0.9% for each subsequent hologram. The total dose for recording all 56 holograms is typically about 2.5 J/cm².

The first hologram set of the third method includes a first hologram recorded using a first recording beam internal angle of +43.519 degrees and a second recording beam internal angle of +163.882 degrees, resulting in a beam difference angle (α) of 120.363 degrees. The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.700 degrees. For each subsequent hologram of first hologram set, the first and second recording beam internal angles are typically changed by amounts that are approximately equal in magnitude to each other, but having opposite signs. For example, for recording a second hologram of the first hologram set, the first recording beam internal angle is changed by +0.351 degree and the second recording beam internal angle is adjusted by −0.355 degree, such that the first recording beam internal angle becomes +43.870 degrees and the second recording beam internal angle +163.527 degrees (α=119.657 degrees). The first and second recording beams of the second hologram are symmetrical about a skew axis having a skew angle of 13.699 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 21 volume holograms of the first hologram set (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of approximately 0.353 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of approximately 0.299 degree for changes in recording beam internal angles from the 20^(th) hologram to the 21^(st) hologram. However, the magnitude of change is approximately the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (21^(st)) hologram of the first hologram set are +49.960 and +157.379 degrees, respectively, and α=107.419 degrees. The first and second recording beams of the 21^(st) hologram are symmetrical about a skew axis having a skew angle of 13.670 degrees.

According to the third method, the first recording beam internal angle of the first hologram set ranges from +43.519 to +49.960 degrees (a range of 6.441 degrees) and the second recording beam internal angle of the first hologram set ranges from +163.882 to +157.379 degrees (a range of 6.503 degrees). For each hologram of the first hologram set, the first recording beam and its respective second recording beam are symmetrical about a skew axis. Thus the internal angle of the first recording beam relative to a skew axis (+29.819 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+150.182 degrees for the first hologram) is equal to 180.0 degrees (29.818°+150.182=) 180.0°. The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles respectively, and the skew angle. The mean skew angle for all holograms of the first hologram set is 13.685 degrees and all skew angles of the first set are within 0.015 degree of the mean. For the first hologram set of the third method of making a skew mirror, first and second recording beam internal angles relative to surface normal of the grating medium and internal angles relative to the skew axis are listed in Table 9.

For many skew mirror applications, all skew angles for a set of holograms are within 2.0 degrees of the mean skew angle for all holograms in the set, in order to achieve adequate reflective performance. In some skew mirror applications, all skew angles for a set of holograms are within 1.0 degree of the hologram set mean skew angle in order to achieve adequate reflective performance. For more demanding applications, all skew angles of a set of holograms are within 0.5 degree of the hologram set mean skew angle in order to achieve adequate reflective performance. For still more demanding applications, all skew angles of a set of holograms are within 0.10 degree of the hologram set mean skew angle in order to achieve adequate reflective performance. For especially demanding applications, all skew angles in a set of holograms are within 0.01 degree of the hologram set mean skew angle.

K_(G) for the holograms of the first hologram set ranges from 4.140×10⁷ rad/m for the first hologram to 3.846×10⁷ rad/m for the 21^(st) hologram, based on n=1.538 for 405 nm light and AK283 photosensitive grating medium, resulting in k=2.386×10⁷ rad/m. The third method can be, but is not necessarily, practiced using the AK283 grating medium having a thickness of 500 μm. Adjacent |ΔK_(G)| for each hologram of the first hologram set is 1.469×10⁵ rad/m. |ΔK_(G)| between the first and 21^(st) holograms is 2.939×10⁶ rad/m. Values for α, K_(G), and |ΔK_(G)| for each of the 21 holograms of the first hologram set of the third method can be found in Table 9.

The second hologram set of the third method includes a first hologram recorded using a first recording beam internal angle of +53.704 degrees and a second recording beam internal angle of +153.696 degrees, resulting in α=99.992 degrees. The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.700 degrees. For recording a second hologram of the second hologram set, the first recording beam internal angle is changed by +0.272 degree and the second recording beam internal angle is adjusted by −0.275 degree, such that the first recording beam internal angle becomes +53.976 degrees and the second recording beam internal angle +153.421 degrees (α=99.445 degrees). The first and second recording beams of the second hologram are symmetrical about a skew axis having a skew angle of 13.699 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 19 volume holograms of the second hologram set (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of approximately 0.274 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of 0.252 degree for changes in recording beam internal angles from the 18^(th) hologram to the 19^(th) hologram. However, the magnitude of change is approximately the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (19^(th)) hologram of the second hologram set are +58.393 and +148.957 degrees, respectively, and α=90.564 degrees. The first and second recording beams of the 19^(th) hologram are symmetrical about a skew axis having a skew angle of 13.675 degrees.

K_(G) for the holograms of the second hologram set ranges from 3.655×10⁷ for the first hologram to 3.391×10⁷ for the 19^(th) hologram (n=1.538; k=2.386×10⁷). Adjacent |ΔK_(G)| for each hologram of the second hologram set is 1.469×10⁵. |ΔK_(G)| between the first and 19^(th) holograms is 2.645×10⁶. Values for α, K_(G), and |ΔK_(G)| for each of the 19 holograms of the second hologram set of the third method can be found in Table 9.

According to the third method, the first recording beam internal angle of the second hologram set ranges from +53.704 to +58.393 degrees (a range of 4.689 degrees) and the second recording beam internal angle of the second hologram set ranges from +153.696 to +148.597 degrees (a range of 4.736 degrees). For each hologram of the second hologram set, the first recording beam and its respective second recording beam are symmetrical about a skew axis. Thus the internal angle of the first recording beam relative to a skew axis (+40.004 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+139.996 degrees for the first hologram) is equal to 180.0 degrees (40.004°+139.996=) 180.0°. The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles respectively, and the skew angle. The mean skew angle for all holograms of the second hologram set is 13.688 degrees and all skew angles of the first set are within 0.013 degree of the mean. For the second hologram set of the third method of making a skew mirror, first and second recording beam internal angles relative to surface normal of the grating medium and internal angles relative to the skew axis are listed in Table 9.

The third hologram set of the third method includes a first hologram recorded using a first recording beam internal angle of +63.696 degrees and a second recording beam internal angle of +143.704 degrees, resulting in α=80.008 degrees. The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.700 degrees. For recording a second hologram of the third hologram set, the first recording beam internal angle is changed by +0.229 degree and the second recording beam internal angle is adjusted by −0.231 degree, such that the first recording beam internal angle becomes +63.925 degrees and the second recording beam internal angle +143.473 degrees (α=79.548 degrees). The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.699 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 16 volume holograms of the third hologram set (i.e., the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of approximately 0.230 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of approximately 0.219 degree for changes in recording beam internal angles from the 15^(th) hologram to the 16^(th) hologram. However, the magnitude of change is approximately the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (16^(th)) hologram of the third hologram set are +67.051 and +140.313 degrees, respectively, and α=73.262 degrees. The first and second recording beams of the 16^(th) hologram are symmetrical about a skew axis having a skew angle of 13.682 degrees.

K_(G) for the holograms of the third hologram set ranges from 3.068×10⁷ for the first hologram to 2.847×10⁷ for the 16^(th) hologram (n=1.538; k=2.386×10⁷). Adjacent |ΔK_(G)| for each hologram of the third hologram set is 1.469×10⁵. |ΔK_(G)| between the first and 16^(th) holograms is 2.204×10⁶. Values for α, K_(G), and |ΔK_(G)| for each of the 16 holograms of the third hologram set of the third method can be found in Table 9.

According to the third method, the first recording beam internal angle of the third hologram set ranges from +63.696 to +67.051 degrees (a range of 3.355 degrees) and the second recording beam internal angle of the third hologram set ranges from +143.704 to +140.313 degrees (a range of 3.391 degrees). For each hologram of the third hologram set, the first recording beam and its respective second recording beam are symmetrical about a skew axis. Thus the internal angle of the first recording beam relative to a skew axis (+49.996 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+130.004 degrees for the first hologram) is equal to 180.0 degrees (49.996°+130.004=) 180.0°. The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles respectively, and the skew angle. The mean skew angle for all holograms of the third hologram set is 13.691 degrees and all skew angles of the first set are within 0.009 degree of the mean. For the third hologram set of the third method of making a skew mirror, first and second recording beam internal angles relative to surface normal of the grating medium and internal angles relative to the skew axis are listed in Table 9.

TABLE 9 RECORDING BEAM ANGLES AND RELATED DATA FOR A THIRD METHOD OF MAKING A SKEW MIRROR Skew Internal Internal Angle Angle of Angle of First Second (internal, First Second Recording Recording in Recording Recording Magnitude Beam Beam degrees, Beam Beam of Angle Angle Angle relative Relative Relative Difference (internal, in degrees, to To Skew To Skew From relative to surface surface Axis Axis Previous α K_(G) # normal) normal) (degrees) (degrees) Hologram (degrees) (×10⁷ rad/m) First Set of Holograms 1 43.519 163.882 13.700 29.819 150.182 120.363 4.140 2 43.870 163.527 13.699 30.171 149.829 0.354 119.657 4.126 3 44.218 163.177 13.697 30.521 149.479 0.351 118.959 4.111 4 44.562 162.830 13.696 30.866 149.134 0.347 118.268 4.096 5 44.903 162.486 13.695 31.208 148.792 0.344 117.583 4.082 6 45.240 162.146 13.693 31.547 148.453 0.340 116.906 4.067 7 45.574 161.809 13.692 31.883 148.117 0.337 116.235 4.052 8 45.905 161.475 13.690 32.215 147.785 0.334 115.570 4.037 9 46.233 161.144 13.689 32.545 147.455 0.331 114.911 4.023 10 46.558 160.816 13.687 32.871 147.129 0.328 114.258 4.008 11 46.880 160.491 13.686 33.195 146.805 0.325 113.611 3.993 12 47.200 160.169 13.684 33.515 146.485 0.322 112.969 3.979 13 47.516 159.849 13.683 33.834 146.166 0.320 112.333 3.964 14 47.830 159.532 13.681 34.149 145.851 0.317 111.702 3.949 15 48.142 159.218 13.680 34.462 145.538 0.314 111.076 3.935 16 48.451 158.905 13.678 34.773 145.227 0.312 110.455 3.920 17 48.757 158.596 13.676 35.081 144.919 0.310 109.838 3.905 18 49.061 158.288 13.675 35.387 144.613 0.307 109.227 3.891 19 49.363 157.983 13.673 35.690 144.310 0.305 108.620 3.876 20 49.663 157.680 13.672 35.991 144.009 0.303 108.017 3.861 21 49.960 157.379 13.670 36.290 143.710 0.301 107.419 3.846 MEAN SKEW ANGLE = 13.685 Second Set of Holograms 1 53.704 153.696 13.700 40.004 139.996 99.992 3.655 2 53.976 153.421 13.699 40.278 139.722 0.275 99.445 3.641 3 54.247 153.148 13.698 40.550 139.450 0.273 98.900 3.626 4 54.517 152.876 13.696 40.820 139.180 0.272 98.359 3.611 5 54.785 152.605 13.695 41.090 138.910 0.270 97.821 3.597 6 55.051 152.336 13.694 41.357 138.643 0.269 97.286 3.582 7 55.316 152.069 13.692 41.624 138.377 0.268 96.753 3.567 8 55.579 151.803 13.691 41.888 138.112 0.266 96.223 3.553 9 55.842 151.538 13.690 42.152 137.848 0.265 95.696 3.538 10 56.102 151.274 13.688 42.414 137.586 0.264 95.172 3.523 11 56.362 151.012 13.687 42.675 137.325 0.262 94.650 3.509 12 56.620 150.751 13.685 42.935 137.065 0.261 94.131 3.494 13 56.877 150.491 13.684 43.193 136.807 0.260 93.614 3.479 14 57.133 150.232 13.683 43.450 136.550 0.259 93.100 3.464 15 57.387 149.975 13.681 43.706 136.294 0.257 92.588 3.450 16 57.640 149.719 13.680 43.961 136.039 0.256 92.079 3.435 17 57.892 149.464 13.678 44.214 135.786 0.255 91.571 3.420 18 58.143 149.210 13.677 44.467 135.533 0.254 91.067 3.406 19 58.393 148.957 13.675 44.718 135.282 0.253 90.564 3.391 MEAN SKEW ANGLE = 13.688 Third Set of Holograms 1 63.696 143.704 13.700 49.996 130.004 80.008 3.068 2 63.925 143.473 13.699 50.226 129.774 0.231 79.548 3.053 3 64.153 143.243 13.698 50.455 129.545 0.230 79.090 3.038 4 64.380 143.013 13.697 50.683 129.317 0.230 78.633 3.024 5 64.607 142.785 13.696 50.911 129.089 0.229 78.178 3.009 6 64.833 142.556 13.694 51.138 128.862 0.228 77.724 2.994 7 65.058 142.329 13.693 51.364 128.636 0.227 77.272 2.980 8 65.282 142.102 13.692 51.590 128.410 0.227 76.821 2.965 9 65.506 141.877 13.691 51.815 128.186 0.226 76.371 2.950 10 65.728 141.651 13.690 52.039 127.961 0.225 75.923 2.935 11 65.951 141.427 13.689 52.262 127.738 0.225 75.476 2.921 12 66.172 141.203 13.687 52.485 127.515 0.224 75.031 2.906 13 66.393 140.979 13.686 52.707 127.293 0.223 74.586 2.891 14 66.613 140.757 13.685 52.928 127.072 0.223 74.144 2.877 15 66.832 140.534 13.683 53.149 126.851 0.222 73.702 2.862 16 67.051 140.313 13.682 53.369 126.631 0.222 73.262 2.847 MEAN SKEW ANGLE = 13.691 A Multicolor Skew Mirror Embodiment

A skew mirror produced by the third method of making a skew mirror can be referred to as a multicolor skew mirror because its grating medium is configured to reflect blue, green, and red light about substantially constant reflective axes. The first hologram set is configured to reflect incident light residing in a blue region of the visible spectrum about substantially constant first reflective axes that differ by at least 2.0 degrees from surface normal of the grating medium. For purposes of the present disclosure, incident light in the blue region of the visible spectrum has a wavelength in the range of 405 nm to 492 nm. The first hologram set is more specifically configured to reflect blue incident light having a wavelength of 463 nm about substantially constant first reflective axes having a mean reflective axis angle of +13.685 degrees, where (i) the blue incident light has internal angles of incidence (relative to surface normal) that range from +8.615 degrees to −8.606 degrees, and (ii) the internal angles of incidence include at least 21 different incidence angles, each of which is separated from all others of the at least 21 different incidence angles by 0.52 degrees or more. In some embodiments, the internal angles of incidence of the blue incident light include at least 4 different incidence angles, each of which is separated from all others of the at least 4 difference incidence angles by 1.0 degrees or more.

The incident light is reflected at an internal angle of reflection (relative to surface normal) ranging from +18.785 degrees to +35.946 degrees, respectively, and the reflected light has the same wavelength as the incident light. Persons skilled in the art recognize that the incident light and its reflection are interchangeable such that where the 463 nm incident light has internal angles of incidence that range from +18.785 degrees to +35.946 degrees, it is reflected about the substantially constant reflective axes at an internal angles of reflection ranging from +8.615 degrees to −8.606 degrees, respectively.

The second hologram set is configured to reflect incident light residing in a green region of the visible spectrum about substantially constant second reflective axes that differ by at least 2.0 degrees from surface normal of the grating medium. For purposes of the present disclosure, incident light in the green region of the visible spectrum has a wavelength in the range of 493 nm to 577 nm. The second hologram set is more specifically configured to reflect green incident light having a wavelength of 522 nm about substantially constant second reflective axes having a mean reflective axis angle of +13.688 degrees, where (i) the green incident light has internal angles of incidence (relative to surface normal) that range from +7.813 degrees to −8.993 degrees, and (ii) the internal angles of incidence include at least 19 different incidence angles, each of which is separated from all others of the at least 19 different incidence angles by 0.60 degrees or more. In some embodiments, the internal angles of incidence of the green incident light include at least 4 different incidence angles, each of which is separated from all others of the at least 4 difference incidence angles by 1.2 degrees or more.

The green incident light is reflected at internal angles of reflection ranging from +19.587 degrees to +36.342 degrees, respectively, and the reflected light has the same wavelength as the incident light. Persons skilled in the art recognize that the incident light and its reflection are interchangeable such that where the 522 nm incident light has an internal angle of incidence that ranges from +19.587 degrees to +36.342 degrees, it is reflected about the substantially constant reflective axes at an internal angle of reflection ranging from +7.813 degrees to −8.993 degrees, respectively.

The third hologram set is configured to reflect incident light residing in a red region of the visible spectrum about substantially constant third reflective axes that differ by at least 2.0 degrees from surface normal of the grating medium. For purposes of the present disclosure, incident light in the red region of the visible spectrum has a wavelength in the range of 610 nm to 780 nm. The third hologram set is more specifically configured to reflect red incident light having a wavelength of 622 nm about substantially constant third reflective axes having a mean reflective axis angle of +13.691 degrees, where (i) the red incident light has internal angles of incidence (relative to surface normal) that range from +10.370 degrees to −8.391 degrees, and (ii) the internal angles of incidence include at least 16 different incidence angles, each of which is separated from others of the at least 16 different incidence angles by 0.74 degrees or more. In some embodiments, the internal angles of incidence of the red incident light include at least 4 different incidence angles, each of which is separated from all others of the at least 4 difference incidence angles by 1.5 degrees or more.

The red incident light is reflected at internal angles of reflection ranging from +17.030 degrees to +35.791 degrees, respectively, and the reflected light has the same wavelength as the incident light. Persons skilled in the art recognize that the red incident light and its reflection are interchangeable such that where the 622 nm incident light has an internal angle of incidence that ranges from +17.030 degrees to +35.791 degrees, it is reflected about the substantially constant reflective axis at internal angles of reflection ranging from +10.370 degrees to −8.391 degrees, respectively.

As described above, the first hologram set is configured to reflect blue incident light having a wavelength of 463 nm about reflective axes that are substantially constant, having a mean reflective axis angle of +13.7 degrees, where the 463 nm light is incident upon the grating medium at multiple internal angles ranging from −8.6 degrees to +8.6 degrees relative to surface normal. The second hologram set is configured to reflect green incident light having a wavelength of 522 nm about reflective axes that are substantially constant, having a mean reflective axis angle of +13.7 degrees, where the 522 nm light is incident upon the grating medium at multiple internal angles of incidence ranging from −9.0 degrees to +7.8 degrees relative to surface normal. The third hologram set is configured to reflect red incident light having a wavelength of 622 nm about reflective axes that are substantially constant, having a mean reflective axis angle of +13.7 degrees, where the 622 nm light is incident upon the grating medium at multiple internal angles ranging from −8.4 degrees to +10.4 degrees relative to surface normal.

Thus the multicolor skew mirror's reflective properties enable it to reflect blue, green, and red incident light about substantially constant reflective axes having a mean reflective axis angle of 13.7 degrees, where the blue, green, and red incident light is incident upon the mirror at internal angles of incidence ranging from −8.4 degrees to +7.8 degrees (a range of 16.2 degrees) relative to surface normal. In embodiments, a skew mirror's reflective properties enable it to reflect blue, green, and red incident light about substantially constant reflective axes, where the blue, green, and red incident light is incident upon the grating medium at multiple internal angles of incidence that span a range of at least 4.0 degrees, or at least 8.0 degrees.

A Multiwavelength Method of Making a Skew Mirror

In a multiwavelength method of making a skew mirror, six volume holograms are recorded in AK233-200 grating medium, with each of the six holograms being recorded using its own unique first and second recording beam internal angles of incidence. In addition, for each of the six volume holograms, wavelengths of the first and second recording beams are adjusted continuously and synchronously from 403 nm to 408 nm, using a variable wavelength laser. Wavelengths of the first and second recording beams are kept equal to each other while recording each of the six volume holograms. Total energy dose delivered in recording the six volume holograms according to the multiwavelength method is typically, but not necessarily, 2.5 J/cm² for first and second recording beam internal angles of incidence for the multiwavelength method of making a skew mirror are provided below in Table 10. A skew mirror made by the multiwavelength method has the same reflective characteristics of the second embodiment skew mirror described above.

TABLE 10 RECORDING BEAM ANGLES FOR A MULTIWAVELENGTH METHOD OF MAKING A SKEW MIRROR First Recording Second Recording Beam Angle of Beam Angle of HOLOGRAM Incidence* Incidence* 1 56.235° 153.001° 2 57.033° 152.203° 3 57.813° 151.423° 4 58.568° 150.668° 5 59.303° 149.933° 6 60.018° 149.218° *internal, relative to grating medium surface normal Other Skew Mirror Embodiments

Embodiments of a skew mirror can be created in a grating medium comprising a volumetric dielectric medium, such as a photosensitive recording medium. Skew mirror embodiments may be formed by constraining a spatial dielectric modulation spectrum as described herein. In an embodiment, dielectric modulation is accomplished holographically by recording an interference pattern of two or more coherent light beams in a photosensitive recording medium. In other embodiments, dielectric modulation can be accomplished by other means.

FIG. 15 illustrates another skew mirror embodiment, a “narcissist's mirror” includes several skew mirrors 1500 whose reflective axes 1561 intersect. A narcissist can sit at the point of convergence and see several images of them self.

Skew Mirror Fabrication

Skew mirrors may be recorded holographically according to an embodiment. Skew mirrors may be recorded holographically or fabricated by with non-holographic means according to embodiments.

Holographic Recording

FIGS. 16A and 16B illustrate additional methods for recording skew mirrors. In FIG. 16A, substantially collimated recording beams are used to illuminate a grating medium to create a desired Δn({right arrow over (k)}) distribution. In one embodiment, illustrated in FIG. 16A, a recording beam pair consisting of a first recording beam 1654A and a second recording beam 1655A at wavelength A illuminate the grating medium 1610 in order to record a first point-like subset of the desired line segment-like Δn({right arrow over (k)}) distribution, e.g., the highest spatial frequency components (the outer tips of Δn({right arrow over (k)}). The angles of incidence θ₁ and θ₂ of a recording apparatus are then adjusted to produce another set of recording beams consisting of another first recording beam 1654B and another second recording beam 1655B, which are also at wavelength λ. The other first and second recording beams 1654B, 1655B illuminate the medium to record a second point-like subset of the desired line segment-like Δn({right arrow over (k)}) distribution. This process is repeated using yet another set of recording beams consisting of yet another first recording beam 1654C and yet another second recording beam 1655C etc . . . , until an entire desired line segment-like Δn({right arrow over (k)}) distribution has been recorded.

In some embodiments, this recording may be made in one continuous exposure wherein θ_(r) and θ_(s) are adjusted continuously and synchronously in order to produce the desired distribution. In other embodiments, separate, discreet exposures where θ_(r) and θ_(s) are fixed during exposure and changed only between exposures are used. Still other embodiments may combine these methods. In some embodiments, the components of Δn({right arrow over (k)}) may be written in an arbitrary order. In some embodiments, intensity may be varied across one or both beams in order to control the spatial diffraction efficiency profile. In some embodiments, a phase control element (e.g., a mirror mounted on a piezo-electric actuator) may be inserted into one or both beam paths in order to control the phase of each exposure. In some embodiments, more than one skew mirror or broadband skew mirror might be recorded into the same medium.

In the case of discreet exposures, the number and angular density of exposures is sufficient to produce a smooth, continuous line segment-like Δn({right arrow over (k)}) distribution. One skilled in the art will readily calculate the angular selectivity of each hologram produced by a discreet exposure using Kogelnik's theory. In one embodiment, exposures are made at angular increments corresponding to a function of this angular selectivity, e.g., at the angular spacing of the full-width-quarter-maximum (FWQM) of the diffraction efficiency peaks. In other embodiments, the angular exposure density might be finer than this in order to assure a smooth final distribution.

The number of FWQM peaks necessary to span the line segment-like Δn({right arrow over (k)}) distribution may be regarded as an equivalent number of holograms, M, required to form the distribution. Accordingly, the maximum possible diffraction efficiency of the resulting skew mirror may be estimated by η=(M/M/#)² where η is the diffraction efficiency, and M/# is a material parameter characterizing the dynamic range of the recording medium. One skilled in the art will readily determine how to refine this estimate according to the geometry of each individual exposure or the overlap of neighboring exposures.

FIG. 16B illustrates an embodiment where a first prism 1659A and a second prism 1659B are incorporated to produce internal beam angles that are not otherwise accessible due to refraction at the grating medium 1610 surface. This method is typically used, for example, to fabricate the skew coupler of FIG. 12B. One skilled in the art will readily perceive how to modify the configurations of FIGS. 13A and 13B to achieve a desired distribution.

In some embodiments, a single recording wavelength A may be chosen to write the entire line segment-like Δn({right arrow over (k)}) distribution. For example, in an embodiment it is possible to write a skew mirror that operates across all visible wavelengths using only a 405 nm laser source. This has an advantage of requiring sufficient recording medium sensitivity at only a single wavelength, as well as an advantage of simplicity. In some embodiments, more than one recording wavelength is used. In still other cases, a continuously-variable wavelength source is used. In one such embodiment, the recording angles θ_(r) and θ_(s) are held constant, and the recording wavelength is instead changed in order to produce the entire line segment-like Δn({right arrow over (k)}) distribution, or a subset thereof.

Other Fabrication Methods

Other methods for producing a skew mirror fall within the scope of the present invention. In one embodiment, for example, a very thick dielectric layer structure is built up using conventional optical coating means. The structure is designed to produce broadband reflectivity within sub-layers, typically by repetition of a conventional broadband reflective coating design. The thick structure is then ground and polished to produce a surface at an oblique angle to the coating layers. The resulting structure typically exhibits mirror-like behavior with respect to a reflective axis substantially defined by the normal of the coating layers rather than the polished surface, and thus constitutes a skew mirror. In some embodiments, atomically-precise manufacturing methods enable fabrication of skew mirrors by composing dielectric structures atom-by-atom without regard to external surfaces.

Non-Flat Mirrors

Skew mirrors may be said to be non-flat in two senses: 1) When the physical shape of the recording medium is not flat; and 2) when the holographic fringes are not planar.

Non-Slab-Like Mirrors

Embodiments of mirrors according to the present invention, including examples of skew mirrors, broadband mirrors, and holographic mirrors, include holograms recorded in medium that is not slab-like in shape. In an example, in an embodiment, a recording layer is cast with a uniform thickness, but on a curved surface. In another example, a non-uniform recording layer (e.g., wedge-shaped) is utilized. In still another example, an arbitrary shape (e.g., spherical) is molded. In these non-slab-like mirror cases, whether the designation “skew mirror” is appropriate depends on the geometry of the relevant surface(s). Non-slab-like holographic mirrors typically exhibit broadband mirror-like properties.

Mirrors with Non-Planar Holographic Fringes

In some embodiments, it is desirable to introduce optical power or other deliberate aberrations into a reflection. This can be accomplished with an embodiment of a skew mirror by locally varying the direction of the reflective axis, for example so that a plane-wave incident beam is reflected to form a spherical-wave reflected beam, as occurs with a conventional parabolic mirror. Such a skew mirror can be fabricated, for instance, by using one converging and one diverging beam in the fabrication method of FIG. 13 and by recording while changing the wavelength instead of the angle of incidence. Such a mirror can also be fabricated by polishing dielectric layers deposited on a non-flat surface, or by using advanced atomically-precise manufacturing methods.

Other Fabrication Embodiments

Some holographic recording system embodiments incorporates mirrors, lenses and prisms to direct first and second recording beams into the grating medium in such a way that translation of the grating medium is not required to record multiple holograms at varying recording beam internal angles, at approximately the same location in the grating medium.

In some embodiments a prism in addition to the coupling prism may be used to fabricate the skew mirror. In some embodiments a variety of coupling prisms and flat pieces of glass may be used. In some embodiments multiple beams, E_(r) _(_) _(N) and E_(s) _(_) _(N), at multiple wavelengths, λ_(N), may be used. In some embodiments multiple wavelengths may be used to fabricate multiple discrete line segment-like Δn({right arrow over (k)}) distributions. In some embodiments multiple wavelengths may be used to fabricate a line segment-like Δn({right arrow over (k)}) distribution that may be continuous or may include closely spaced sections. In some embodiments the incident angle of the signal and/or reference beam may be adjusted to compensate for shrinkage of the sample material. In some embodiments the sample may be rotated to compensate for shrinkage of the sample material. In some embodiments the wavelength may be changed to compensate for shrinkage of the sample material.

Alternative Embodiments and Variations

The various embodiments and variations thereof, illustrated in the accompanying Figures and/or described above, are merely exemplary and are not meant to limit the scope of the invention. It is to be appreciated that numerous other variations of the invention have been contemplated, as would be obvious to one of ordinary skill in the art, given the benefit of this disclosure. All variations of the invention that read upon appended claims are intended and contemplated to be within the scope of the invention.

Terminology

The terms and phrases as indicated in quotation marks (“ ”) in this section are intended to have the meaning ascribed to them in this Terminology section applied to them throughout this document, including in the claims, unless clearly indicated otherwise in context. Further, as applicable, the stated definitions are to apply, regardless of the word or phrase's case, to the singular and plural variations of the defined word or phrase.

References in the specification to “one embodiment,” “an embodiment,” “another embodiment,” “a preferred embodiment,” “an alternative embodiment,” “one variation,” “a variation,” and similar phrases mean that a particular feature, structure, or characteristic described in connection with the embodiment or variation, is included in at least an embodiment or variation of the invention. The phrase “in one embodiment,” “in one variation” or similar phrases, as used in various places in the specification, are not necessarily meant to refer to the same embodiment or the same variation.

The term “approximately,” as used in this specification and appended claims, refers to plus or minus 10% of the value given.

The term “about,” as used in this specification and appended claims, refers to plus or minus 20% of the value given.

The term “generally,” as used in this specification and appended claims, mean mostly, or for the most part.

The term “principally,” as used in this specification and appended claims with respect to reflected light, refers to light reflected by a grating structure. Light that is principally reflected at a recited angle includes more light than is reflected at any other angle (excluding surface reflections). Light that is principally reflected about a recited reflective axis includes more reflected light than is reflected about any other reflective axis (excluding surface reflections). Light reflected by a device surface is not included when considering principally reflected light.

The term “reflective axis,” as used in this specification and appended claims, refers to an axis that bisects an angle of an incident light ray relative to its reflected light ray. The incident light ray, reflective axis, and reflected light ray all reside in one common plane, which can be referred to as a plane of incidence. The plane of incidence for a skew mirror need not be coincident with surface normal, although it can be. The magnitude of an angle of incidence of the incident light ray relative to the reflective axis is equal to the magnitude of an angle of reflection of the reflected light ray relative to the reflective axis. For purposes of the foregoing definition of “reflective axis,” and angles are internal angles. For conventional dielectric and metal mirrors, the reflective axis is coincident with surface normal, i.e. the reflective axis is perpendicular to the mirror surface, as is the plane of incidence. Conversely, embodiments of skew mirrors according to the present invention may have a reflective axis that differs from surface normal, or may have a reflective axis that is coincident with surface normal. Whether or not a skew mirror's reflective axis is coincident with surface normal is independent of whether or not the skew mirror's plane of incidence is coincident with surface normal. Angles of incidence and angles of reflection are usually, but not necessarily, determined empirically, with multiple measurements (generally three or more) typically used to generate a mean value.

The term “reflection” and similar terms are used in this disclosure in some cases where “diffraction” might ordinarily be considered an appropriate term. This use of “reflection” is consistent with mirror-like properties exhibited by skew mirrors and helps avoid potentially confusing terminology. For example, where a grating structure is said to be configured to “reflect” incident light, a conventional artisan might prefer to say the grating structure is configured to “diffract” incident light, since grating structures are generally thought to act on light by diffraction. However, such use of the term “diffract” would result in expressions such as “incident light is diffracted about substantially constant reflective axes,” which could be confusing. Accordingly, where incident light is said to be “reflected” by a grating structure, persons of ordinary skill in art, given the benefit of this disclosure, will recognize that the grating structure is in fact “reflecting” the light by a diffractive mechanism. Such use of “reflect” is not without precedent in optics, as conventional dielectric mirrors are generally said to “reflect” light despite the predominant role diffraction plays in such reflection. Artisans of ordinary skill thus recognize that most “reflection” includes characteristics of diffraction, and “reflection” by a skew mirror or components thereof also includes diffraction.

The terms “angle interval” and “angle intervals,” as used in this specification and appended claims, refer to angular spacing between multiple light beams incident upon a skew mirror within a recited range of angles of incidence.

The terms “hologram” and “holographic grating,” as used in this specification and appended claims, refer to a recording of an interference pattern generated by interference between multiple intersecting light beams. A hologram or holographic grating is an example of a grating structure.

While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.

The above-described embodiments can be implemented in any of numerous ways. For example, embodiments of designing and making the technology disclosed herein may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.

Further, it should be appreciated that a computer may be embodied in any of a number of forms, such as a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computer may be embedded in a device not generally regarded as a computer but with suitable processing capabilities, including a Personal Digital Assistant (PDA), a smart phone or any other suitable portable or fixed electronic device.

Also, a computer may have one or more input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can be used to provide a user interface include printers or display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that can be used for a user interface include keyboards, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computer may receive input information through speech recognition or in other audible format.

Such computers may be interconnected by one or more networks in any suitable form, including a local area network or a wide area network, such as an enterprise network, and intelligent network (IN) or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks.

The various methods or processes (e.g., of designing and making the coupling structures and diffractive optical elements disclosed above) outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine.

In this respect, various inventive concepts may be embodied as a computer readable storage medium (or multiple computer readable storage media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other non-transitory medium or tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments of the invention discussed above. The computer readable medium or media can be transportable, such that the program or programs stored thereon can be loaded onto one or more different computers or other processors to implement various aspects of the present invention as discussed above.

The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as discussed above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present invention need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present invention.

Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments.

Also, data structures may be stored in computer-readable media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements.

Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.

The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 221.03. 

We claim:
 1. Apparatus comprising: a grating medium; a grating structure residing in the grating medium, wherein: the grating structure is configured to reflect first incident light, the first incident light being incident upon the grating medium at a specific site and having a first wavelength and a first internal angle of incidence relative to a grating medium surface normal; the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having the first wavelength and a first internal angle of reflection relative to the surface normal; the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal; the grating structure is further configured to reflect second incident light, the second incident light being incident on the grating medium at the specific site and having a second wavelength and a second internal angle of incidence relative to the surface normal; the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having the second wavelength and a second internal angle of reflection relative to the surface normal; the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal; the first internal angle of incidence is the same as the second internal angle of incidence; the first reflective axis is non-zero relative to the surface normal; the first wavelength differs from the second wavelength; and the first reflective axis angle differs from the second reflective axis angle.
 2. The apparatus of claim 1, wherein the first reflective axis angle differs from the second reflective axis angle by 0.25 degrees or less.
 3. The apparatus of claim 2, wherein the first incident light is offset from the first reflective axis by at least 1.0 degree.
 4. The apparatus of claim 1, wherein the first wavelength differs from the second wavelength by a wave fraction of at least 0.005.
 5. The apparatus of claim 1, wherein: the grating structure comprises a plurality of volume holograms; each of the volume holograms in the plurality of volume holograms spatially overlaps at least one other volume holograms in the plurality of volume holograms; and the grating medium is at least 70 μm thick.
 6. The apparatus of claim 5, wherein: the plurality of volume holograms includes at least four holograms; and each of the volume holograms in the plurality of volume holograms at least partially spatially overlaps all others of the plurality of volume holograms.
 7. The apparatus of claim 6, wherein adjacent |ΔK_(G)| for the at least four holograms has a mean value that resides between 5.0×10³ and 1.0×10⁷ radians per meter (rad/m).
 8. The apparatus of claim 1, wherein: the grating structure comprises at least 9 volume holograms; each of the at least 9 volume holograms at least partially spatially overlaps at least one other of the at least 9 volume holograms; and the grating medium is at least 200 μm thick.
 9. The apparatus of claim 1, wherein the first reflective axis differs from the surface normal by at least 4.0 degrees.
 10. The apparatus of claim 9, wherein the first reflective axis differs from the surface normal by at least 9.0 degrees.
 11. A method of making the apparatus of claim 1, the method comprising: creating the grating structure by recording multiple volume holograms in the grating medium, wherein: each of the multiple volume holograms is recorded using a first recording beam and a second recording beam, each of the first and second recording beams including a collimated, plane wave beam, and the first recording beam having the same wavelength as the second recording beam; each of the multiple volume holograms is recorded with the first recording beam being incident upon the grating medium at a unique first recording beam internal angle relative to the surface normal and the second recording beam being incident upon the grating medium at a unique second recording beam internal angle relative to the surface normal; each of the multiple volume holograms is recorded with the first recording beam and the second recording beam being symmetrical about a skew axis; each of the multiple volume holograms at least partially spatially overlaps at least one other of the multiple holograms; the skew axes of the multiple volume holograms have substantially constant skew angles relative to the surface normal; and the skew axes of the multiple volume holograms have a mean skew angle that is substantially identical to both the first reflective axis angle and the second reflective axis angle.
 12. The method of claim 11, wherein each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.
 13. A method of making an apparatus, the method comprising: creating a grating structure in a grating medium by recording multiple volume holograms in the grating medium, wherein: each of the multiple volume holograms is recorded using a first recording beam and a second recording beam, each of the first and second recording beams including a collimated, plane wave beam, and the first recording beam having a same wavelength as the second recording beam; each of the multiple volume holograms is recorded with the first recording beam being incident upon the grating medium at a unique first recording beam internal angle relative to a surface normal of the grating medium and the second recording beam being incident upon the grating medium at a unique second recording beam internal angle relative to the surface normal; each of the multiple volume holograms is recorded with the first recording beam and the second recording beam being symmetrical about a skew axis, the skew axis having a skew axis angle relative to the surface normal; each of the multiple volume holograms at least partially spatially overlaps at least one other of the multiple holograms; and the skew axes of the multiple volume holograms are substantially constant and have a non-zero mean skew axis angle relative to the surface normal.
 14. The method of claim 13, wherein each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.
 15. The method of claim 14, wherein: the multiple volume holograms includes at least 9 holograms; and all of the unique first recording beam internal angles collectively span a range of at least 6.4 degrees.
 16. The method of claim 15, wherein: adjacent |ΔK_(G)| for the at least 9 holograms has a mean value that resides between 5.0×10³ and 1.0×10⁷ rad/m.
 17. The method of claim 15, wherein: adjacent |ΔK_(G)| for the at least 9 holograms has a mean value that resides between 1.0×10⁴ and 1.0×10⁶ rad/m.
 18. The method of claim 13, wherein: the grating structure is configured to reflect first incident light, the first incident light being incident upon the grating medium at a specific site and having a first wavelength and a first internal angle of incidence relative to grating medium surface normal; the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having the first wavelength and a first internal angle of reflection relative to the surface normal; the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal; the grating structure is further configured to reflect second incident light, the second incident light being incident on the grating medium at the specific site and having a second wavelength and a second internal angle of incidence relative to the surface normal; the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having the second wavelength and a second internal angle of reflection relative to the surface normal; the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal; the first internal angle of incidence is the same as the second internal angle of incidence; both the first reflective axis angle and the second reflective axis angle are substantially identical to the skew axis angle; and the first wavelength differs from the second wavelength.
 19. Apparatus comprising: a grating medium; a grating structure residing in the grating medium, wherein: the grating structure is configured to reflect first incident light, the first incident light being incident upon the grating medium at a specific site and having a first internal angle of incidence relative to a grating medium surface normal; the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having a first internal angle of reflection relative to the surface normal; the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal; the grating structure is further configured to reflect second incident light, the second incident light being incident on the grating medium at the specific site and having a second internal angle of incidence relative to the surface normal; the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having a second internal angle of reflection relative to the surface normal; the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal; the first incident light, the first reflected light, the second incident light, and the second reflected light have the same wavelength as each other; the first internal angle of incidence differs from the second internal angle of incidence; the first reflective axis differs from the surface normal by a non-zero angle; and the first reflective axis angle differs from the second reflective axis angle.
 20. The apparatus of claim 19, wherein: the first reflective axis differs from surface normal by at least 1.0 degree.
 21. The apparatus of claim 19, wherein: the first reflective axis angle differs from the second reflective axis angle by 0.25 degrees or less.
 22. The apparatus of claim 19, wherein: each of the first incident light and the second incident light are offset from the first reflective axis by an internal angle of at least 1.0 degree.
 23. The apparatus of claim 19, wherein: the first internal angle of incidence differs from the second internal angle of incidence by at least 6.6 degrees.
 24. Apparatus comprising: a grating medium; a grating structure residing in the grating medium, wherein: the grating structure includes a first hologram set, a second hologram set, and a third hologram set, each of which comprises a plurality of volume holograms; the first hologram set is configured to reflect blue incident light about a substantially constant first reflective axis having a first mean reflective axis angle relative to a surface normal of the grating medium; the second hologram set is configured to reflect green incident light about a second reflective axis having a mean second reflective axis angle relative to the surface normal; the third hologram set is configured to reflect red incident light about a third reflective axis having a third mean reflective axis angle relative to the surface normal; each of the first, second, and third mean reflective axes angles is non-zero; each of the first, second, and third hologram sets includes at least four volume holograms; and within each of the first, second, and third hologram sets, each of the plurality of volume holograms at least partially spatially overlaps at least one other volume hologram of the plurality of volume holograms in that hologram set.
 25. The apparatus of claim 24, wherein the grating medium is at least 200 μm thick.
 26. The apparatus of claim 24, wherein the grating medium is at least 70 μm thick.
 27. The apparatus of claim 24, wherein: each of the blue, green, and red incident light is incident upon the grating medium at multiple internal angles of incidence that span a range of at least 4.0 degrees.
 28. The apparatus of claim 27, wherein; within each of the first, second, and third hologram sets, each of the volume holograms in the plurality of volume holograms at least partially spatially overlaps all other volume holograms in the plurality of volume holograms in that hologram set.
 29. The apparatus of claim 24, wherein: the first mean reflective axis angle is within 2.0 degrees of the second and third mean reflective axis angles and the third mean reflective axis angle is within 2.0 degrees of the second mean reflective axis angle.
 30. The apparatus of claim 24, wherein adjacent |ΔK_(G)| for each of the first, second, and third hologram sets has a mean value that resides between 5.0×10³ and 1.0×10⁷ rad/m.
 31. The apparatus of claim 24, wherein each of the first, second, and third hologram sets includes at least five volume holograms.
 32. A method of using an apparatus comprising: projecting light at the apparatus, wherein: the apparatus comprises a grating medium within which resides a grating structure; the grating medium is at least 70 μm thick; the light includes first incident light, the first incident light being incident upon the grating medium at a specific site and having a first wavelength and a first internal angle of incidence relative to a grating medium surface normal; the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having the first wavelength and a first internal angle of reflection relative to the grating medium surface normal; the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the grating medium surface normal; the light further includes second incident light, the second incident light being incident on the grating medium at the specific site and having a second wavelength and a second internal angle of incidence relative to the grating medium surface normal; the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having the second wavelength and a second internal angle of reflection relative to the grating medium surface normal; the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the grating medium surface normal; the first internal angle of incidence is the same as the second internal angle of incidence; the first reflective axis differs from the grating medium surface normal by a non-zero angle; the first wavelength differs from the second wavelength; and the first reflective axis angle differs from the second reflective axis angle.
 33. The method of claim 32, wherein the first reflective axis differs from the surface normal by at least 1.0 degree.
 34. The method of claim 33, wherein the first incident light is offset from the first reflective axis by an internal angle of at least 5.0 degrees.
 35. The method of claim 32, wherein: the grating medium is at least 200 μm thick; the grating structure comprises multiple volume holograms; and each of the multiple volume holograms at least partially spatially overlaps at least one other of the multiple volume holograms.
 36. The method of claim 35, wherein: the multiple volume holograms includes at least 6 holograms; and each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.
 37. The method of claim 32, wherein: the grating structure comprises at least 9 volume holograms; each of the at least 9 volume holograms at least partially spatially overlaps at least one other of the at least 9 volume holograms; and adjacent |ΔK_(G)| for the at least 9 holograms has a mean value that resides between 1.0×10⁴ and 1.0×10⁶ radians per meter (rad/m).
 38. The method of claim 37, wherein: each of the at least 9 volume holograms at least partially spatially overlaps all others of the at least 9 volume holograms. 